Part B
END-OF-CHAPTER
SOLUTIONS
Fundamentals of Investments, 5th edition
Jordan and Miller
, Chapter 1
A Brief History of Risk and Return
Concept Questions
1. For both risk and return, increasing order is b, c, a, d. On average, the higher the risk of an
investment, the higher is its expected return.
2. Since the price didn’t change, the capital gains yield was zero. If the total return was four percent,
then the dividend yield must be four percent.
3. It is impossible to lose more than –100 percent of your investment. Therefore, return distributions
are cut off on the lower tail at –100 percent; if returns were truly normally distributed, you could lose
much more.
4. 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.
5. Blume’s formula uses the arithmetic and geometric returns along with the number of observations to
approximate a holding period return. When predicting a holding period return, the arithmetic return
will tend to be too high and the geometric return will tend to be too low. Blume’s formula
statistically adjusts these returns for different holding period expected returns.
6. T-bill rates were highest in the early eighties since inflation at the time was relatively high. As we
discuss in our chapter on interest rates, rates on T-bills will almost always be slightly higher than the
rate of inflation.
7. 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.
8. Returns, risk premiums, and volatility would all be lower than we estimated because aftertax returns
are smaller than pretax returns.
9. We have seen that T-bills barely kept up with inflation before taxes. After taxes, investors in T-bills
actually lost ground (assuming anything other than a very low tax rate). Thus, an all T-bill strategy
will probably lose money in real dollars for a taxable investor.
10. It is important not to lose sight of the fact that the results we have discussed cover over 70 years,
well beyond the investing lifetime for most of us. There have been extended periods during which
small stocks have done terribly. Thus, one reason most investors will choose not to pursue a 100
percent stock (particularly small-cap stocks) strategy is that many investors have relatively short
horizons, and high volatility investments may be very inappropriate in such cases. There are other
reasons, but we will defer discussion of these to later chapters.
, CHAPTER 1 B – 2
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.
Core Questions
1. Total dollar return = 100($82 – 73 + 0.88) = $988.00
Whether you choose to sell the stock or not does not affect the gain or loss for the year, your stock is
worth what it would bring if you sold it. Whether you choose to do so or not is irrelevant (ignoring
commissions and taxes).
2. Capital gains yield = ($82 – 73)/$73 = 12.33%
Dividend yield = $0.88/$73 = 1.21%
Total rate of return = 12.33% + 1.21% = 13.53%
3. Dollar return = 750($68.40 – 73 + 0.88) = –$2,790.00
Capital gains yield = ($68.40 – 73)/$73 = –6.30%
Dividend yield = $0.88/$73 = 1.21%
Total rate of return = –6.30% + 1.21% = –5.10%
4. a. average return = 5.8%, average risk premium = 2.0%
b. average return = 3.8%, average risk premium = 0%
c. average return = 12.3%, average risk premium = 8.5%
d. average return = 17.4%, average risk premium = 13.6%
5. Cherry average return = (7% + 21% – 8% – 5% + 31%) / 5 = 9.20%
Straw average return = (12% + 34% – 11% – 7% + 41%) / 5 = 13.80%
6. Stock A: RA = (0.17 + 0.09 – 0.14 + 0.24 + 0.13)/5 = 0. = 9.80%
Var = 1/4[(.17 – .098)2 + (.09 – .098)2 + (–.14 – .098)2 + (.24 – .098)2 + (.13 – .098)2] = 0.02077
Standard deviation = (0.02077)1/2 = 0.1441 or 14.41%
Stock B: RB = (0.21 + 0.07 – 0.10 + 0.26 + 0.15)/5 = 0. = 11.80%
Var = 1/4[(.21 – .118)2 + (.07 – .118)2 + (–.10 – .118)2 + (.26 – .118)2 + (.15 – .118)2] = 0.01987
Standard deviation = (0.01987)1/2 = 0.1410 or 14.10%
7. The capital gains yield is ($59 – 65)/$65 = –.0923 or –9.23% (notice the negative sign). With a
dividend yield of 1.7 percent, the total return is –7.53%.
8. Geometric return = [(1 + .16)(1 – .19)(1 + .04)(1 + .34)(1 + .22)] (1/5) – 1 = .0982 or 9.82%
9. Arithmetic return = (.31 + .09 + .17 –.03 – .21 + .37) / 6 = .1167 or 11.67%
Geometric return = [(1 + .31)(1 + .09)(1 + .17)(1 – .03)(1 – .21)(1 + .37)](1/6) – 1 = .0982 or 9.82%
, B – 3 SOLUTIONS
Intermediate Questions
10. That’s plus or minus one standard deviation, so about two-thirds of the time or two years out of
three. In one year out of three, you will be outside this range, implying that you will be below it one
year out of six and above it one year out of six.
11. You lose money if you have a negative return. With a 7 percent expected return and a 3.5 percent
standard deviation, a zero return is two standard deviations below the average. The odds of being
outside (above or below) two standard deviations are 5 percent; the odds of being below are half
that, or 2.5 percent. (It’s actually 2.28 percent.) You should expect to lose money only 2.5 years out
of every 100. It’s a pretty safe investment.
12. The average return is 5.8 percent, with a standard deviation of 9.2 percent, so Prob( Return < –3.4 or
Return > 15.0 ) ≈ 1/3, but we are only interested in one tail; Prob( Return < –3.4) ≈ 1/6, which is
half of 1/3 .
95%: 5.8 ± 2σ = 5.8 ± 2(9.2) = –12.6% to 24.2%
99%: 5.8 ± 3σ = 5.8 ± 3(9.2) = –21.8% to 33.4%
13. Expected return = 17.4% ; σ = 32.7%. Doubling your money is a 100% return, so if the return
distribution is normal, Z = (100 – 17.4)/32.7 = 2.53 standard deviations; this is in-between two and
three standard deviations, so the probability is small, somewhere between .5% and 2.5% (why?).
Referring to the nearest Z table, the actual probability is = 0.577%, or less than once every 100
years. Tripling your money would be Z = (200 – 17.4)/ 32.7 = 5.58 standard deviations; this
corresponds to a probability of (much) less than 0.5%, or once every 200 years. (The actual answer
is less than once every 1 million years, so don’t hold your breath.)
14. Year Common stocks 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%
1796 23.93% 5.07% 18.86%
1977 –7.16% 5.45% –12.61%
12.84% 31.67% –18.83%
a. Annual risk premium = Common stock return – T-bill return (see table above).
b. Average returns: Common stocks = 12. = 2.57% ; T-bills = 31. = 6.33%;
Risk premium = –18. = –3.77%
c. Common stocks: Var = 1/4[ (–.1469 – .0257)2 + (–.2647 – .0257)2 + (.3723 – .0257)2 +
(.2393 – .0257)2 + (–.0716 – .0257)2] = 0.072337
Standard deviation = (0.072337)1/2 = 0.2690 = 26.90%
T-bills: Var = 1/4[(.0729 – .0633)2 + (.0799 – .0633)2 + (.0587 – .0633)2 + (.0507–.0633)2 +
(.0545 – .0633)2] = 0.0001565
Standard deviation = (0.000156)1/2 = 0.0125 = 1.25%
Risk premium: Var = 1/4[(–.2198 – (–.0377))2 + (–.3446 – (–.0377))2 + (.3136 – (–.0377))2 +
(.1886 – (–.0377))2 + (–.1261 – (–.0377))2] = 0.077446
Standard deviation = (0.077446)1/2 = 0.2783 = 27.83%
END-OF-CHAPTER
SOLUTIONS
Fundamentals of Investments, 5th edition
Jordan and Miller
, Chapter 1
A Brief History of Risk and Return
Concept Questions
1. For both risk and return, increasing order is b, c, a, d. On average, the higher the risk of an
investment, the higher is its expected return.
2. Since the price didn’t change, the capital gains yield was zero. If the total return was four percent,
then the dividend yield must be four percent.
3. It is impossible to lose more than –100 percent of your investment. Therefore, return distributions
are cut off on the lower tail at –100 percent; if returns were truly normally distributed, you could lose
much more.
4. 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.
5. Blume’s formula uses the arithmetic and geometric returns along with the number of observations to
approximate a holding period return. When predicting a holding period return, the arithmetic return
will tend to be too high and the geometric return will tend to be too low. Blume’s formula
statistically adjusts these returns for different holding period expected returns.
6. T-bill rates were highest in the early eighties since inflation at the time was relatively high. As we
discuss in our chapter on interest rates, rates on T-bills will almost always be slightly higher than the
rate of inflation.
7. 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.
8. Returns, risk premiums, and volatility would all be lower than we estimated because aftertax returns
are smaller than pretax returns.
9. We have seen that T-bills barely kept up with inflation before taxes. After taxes, investors in T-bills
actually lost ground (assuming anything other than a very low tax rate). Thus, an all T-bill strategy
will probably lose money in real dollars for a taxable investor.
10. It is important not to lose sight of the fact that the results we have discussed cover over 70 years,
well beyond the investing lifetime for most of us. There have been extended periods during which
small stocks have done terribly. Thus, one reason most investors will choose not to pursue a 100
percent stock (particularly small-cap stocks) strategy is that many investors have relatively short
horizons, and high volatility investments may be very inappropriate in such cases. There are other
reasons, but we will defer discussion of these to later chapters.
, CHAPTER 1 B – 2
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.
Core Questions
1. Total dollar return = 100($82 – 73 + 0.88) = $988.00
Whether you choose to sell the stock or not does not affect the gain or loss for the year, your stock is
worth what it would bring if you sold it. Whether you choose to do so or not is irrelevant (ignoring
commissions and taxes).
2. Capital gains yield = ($82 – 73)/$73 = 12.33%
Dividend yield = $0.88/$73 = 1.21%
Total rate of return = 12.33% + 1.21% = 13.53%
3. Dollar return = 750($68.40 – 73 + 0.88) = –$2,790.00
Capital gains yield = ($68.40 – 73)/$73 = –6.30%
Dividend yield = $0.88/$73 = 1.21%
Total rate of return = –6.30% + 1.21% = –5.10%
4. a. average return = 5.8%, average risk premium = 2.0%
b. average return = 3.8%, average risk premium = 0%
c. average return = 12.3%, average risk premium = 8.5%
d. average return = 17.4%, average risk premium = 13.6%
5. Cherry average return = (7% + 21% – 8% – 5% + 31%) / 5 = 9.20%
Straw average return = (12% + 34% – 11% – 7% + 41%) / 5 = 13.80%
6. Stock A: RA = (0.17 + 0.09 – 0.14 + 0.24 + 0.13)/5 = 0. = 9.80%
Var = 1/4[(.17 – .098)2 + (.09 – .098)2 + (–.14 – .098)2 + (.24 – .098)2 + (.13 – .098)2] = 0.02077
Standard deviation = (0.02077)1/2 = 0.1441 or 14.41%
Stock B: RB = (0.21 + 0.07 – 0.10 + 0.26 + 0.15)/5 = 0. = 11.80%
Var = 1/4[(.21 – .118)2 + (.07 – .118)2 + (–.10 – .118)2 + (.26 – .118)2 + (.15 – .118)2] = 0.01987
Standard deviation = (0.01987)1/2 = 0.1410 or 14.10%
7. The capital gains yield is ($59 – 65)/$65 = –.0923 or –9.23% (notice the negative sign). With a
dividend yield of 1.7 percent, the total return is –7.53%.
8. Geometric return = [(1 + .16)(1 – .19)(1 + .04)(1 + .34)(1 + .22)] (1/5) – 1 = .0982 or 9.82%
9. Arithmetic return = (.31 + .09 + .17 –.03 – .21 + .37) / 6 = .1167 or 11.67%
Geometric return = [(1 + .31)(1 + .09)(1 + .17)(1 – .03)(1 – .21)(1 + .37)](1/6) – 1 = .0982 or 9.82%
, B – 3 SOLUTIONS
Intermediate Questions
10. That’s plus or minus one standard deviation, so about two-thirds of the time or two years out of
three. In one year out of three, you will be outside this range, implying that you will be below it one
year out of six and above it one year out of six.
11. You lose money if you have a negative return. With a 7 percent expected return and a 3.5 percent
standard deviation, a zero return is two standard deviations below the average. The odds of being
outside (above or below) two standard deviations are 5 percent; the odds of being below are half
that, or 2.5 percent. (It’s actually 2.28 percent.) You should expect to lose money only 2.5 years out
of every 100. It’s a pretty safe investment.
12. The average return is 5.8 percent, with a standard deviation of 9.2 percent, so Prob( Return < –3.4 or
Return > 15.0 ) ≈ 1/3, but we are only interested in one tail; Prob( Return < –3.4) ≈ 1/6, which is
half of 1/3 .
95%: 5.8 ± 2σ = 5.8 ± 2(9.2) = –12.6% to 24.2%
99%: 5.8 ± 3σ = 5.8 ± 3(9.2) = –21.8% to 33.4%
13. Expected return = 17.4% ; σ = 32.7%. Doubling your money is a 100% return, so if the return
distribution is normal, Z = (100 – 17.4)/32.7 = 2.53 standard deviations; this is in-between two and
three standard deviations, so the probability is small, somewhere between .5% and 2.5% (why?).
Referring to the nearest Z table, the actual probability is = 0.577%, or less than once every 100
years. Tripling your money would be Z = (200 – 17.4)/ 32.7 = 5.58 standard deviations; this
corresponds to a probability of (much) less than 0.5%, or once every 200 years. (The actual answer
is less than once every 1 million years, so don’t hold your breath.)
14. Year Common stocks 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%
1796 23.93% 5.07% 18.86%
1977 –7.16% 5.45% –12.61%
12.84% 31.67% –18.83%
a. Annual risk premium = Common stock return – T-bill return (see table above).
b. Average returns: Common stocks = 12. = 2.57% ; T-bills = 31. = 6.33%;
Risk premium = –18. = –3.77%
c. Common stocks: Var = 1/4[ (–.1469 – .0257)2 + (–.2647 – .0257)2 + (.3723 – .0257)2 +
(.2393 – .0257)2 + (–.0716 – .0257)2] = 0.072337
Standard deviation = (0.072337)1/2 = 0.2690 = 26.90%
T-bills: Var = 1/4[(.0729 – .0633)2 + (.0799 – .0633)2 + (.0587 – .0633)2 + (.0507–.0633)2 +
(.0545 – .0633)2] = 0.0001565
Standard deviation = (0.000156)1/2 = 0.0125 = 1.25%
Risk premium: Var = 1/4[(–.2198 – (–.0377))2 + (–.3446 – (–.0377))2 + (.3136 – (–.0377))2 +
(.1886 – (–.0377))2 + (–.1261 – (–.0377))2] = 0.077446
Standard deviation = (0.077446)1/2 = 0.2783 = 27.83%
