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 and for
changes in the base used for each year’s calculation of returns. 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 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 expected 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
80 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.
,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($41 – 37 + 0.28) = $428.00
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 = ($41 – 37) / $37 = 10.81%
Dividend yield = $0.28 / $37 = 0.76%
Total rate of return = 10.81% + 0.76% = 11.57%
3. Dollar return = 500($34 – 37 + 0.28) = –$1,360
Capital gains yield = ($34 – 37) / $37 = –8.11%
Dividend yield = $0.28 / $37 = 0.76%
Total rate of return = –8.11% + 0.76% = –7.35%
4. a. average return = 6.2%, average risk premium = 2.5%
b. average return = 3.7%, average risk premium = 0%
c. average return = 11.7%, average risk premium = 8.0%
d. average return = 17.5%, average risk premium = 13.8%
, 5. Cherry average return = (17% + 11% – 2% + 3% + 14%) / 5 = 8.60%
Straw average return = (16% + 18% – 6% + 1% + 22%) / 5 = 10.20%
6. Cherry: RA = 8.60%
Var = 1/4[(.17 – .086)2 + (.11 – .086)2 + (–.02 – .086)2 + (.03 – .086)2 + (.14 – .086)2]
= .00623
Standard deviation = (0.00623)1/2 = .0789, or 7.89%
Straw: RB = 10.20%
Var = 1/4[(.16 – .102)2 + (.18 – .102)2 + (–.06 – .102)2 + (.01 – .102)2 + (.22 – .102)2]
= .01452
Standard deviation = (0.01452)1/2 = .1205, or 12.05%
7. The capital gains yield is ($59 – 65)/$65 = –.0923, or –9.23% (notice the negative sign).
With a dividend yield of 1.2 percent, the total return is –8.03%.
8. Geometric return = [(1 + .17)(1 + .11)(1 - .02)(1 + .03)(1 + .14)] (1/5) – 1 = .0837, or
8.37%
9. Arithmetic return = (.21 + .12 + .07 –.13 – .04 + .26) / 6 = .0817, or 8.17%
Geometric return = [(1 + .21)(1 + .12)(1 + .07)(1 – .13)(1 – .04)(1 + .26)](1/6) – 1 = .0730,
or 7.30%
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 and for
changes in the base used for each year’s calculation of returns. 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 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 expected 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
80 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.
,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($41 – 37 + 0.28) = $428.00
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 = ($41 – 37) / $37 = 10.81%
Dividend yield = $0.28 / $37 = 0.76%
Total rate of return = 10.81% + 0.76% = 11.57%
3. Dollar return = 500($34 – 37 + 0.28) = –$1,360
Capital gains yield = ($34 – 37) / $37 = –8.11%
Dividend yield = $0.28 / $37 = 0.76%
Total rate of return = –8.11% + 0.76% = –7.35%
4. a. average return = 6.2%, average risk premium = 2.5%
b. average return = 3.7%, average risk premium = 0%
c. average return = 11.7%, average risk premium = 8.0%
d. average return = 17.5%, average risk premium = 13.8%
, 5. Cherry average return = (17% + 11% – 2% + 3% + 14%) / 5 = 8.60%
Straw average return = (16% + 18% – 6% + 1% + 22%) / 5 = 10.20%
6. Cherry: RA = 8.60%
Var = 1/4[(.17 – .086)2 + (.11 – .086)2 + (–.02 – .086)2 + (.03 – .086)2 + (.14 – .086)2]
= .00623
Standard deviation = (0.00623)1/2 = .0789, or 7.89%
Straw: RB = 10.20%
Var = 1/4[(.16 – .102)2 + (.18 – .102)2 + (–.06 – .102)2 + (.01 – .102)2 + (.22 – .102)2]
= .01452
Standard deviation = (0.01452)1/2 = .1205, or 12.05%
7. The capital gains yield is ($59 – 65)/$65 = –.0923, or –9.23% (notice the negative sign).
With a dividend yield of 1.2 percent, the total return is –8.03%.
8. Geometric return = [(1 + .17)(1 + .11)(1 - .02)(1 + .03)(1 + .14)] (1/5) – 1 = .0837, or
8.37%
9. Arithmetic return = (.21 + .12 + .07 –.13 – .04 + .26) / 6 = .0817, or 8.17%
Geometric return = [(1 + .21)(1 + .12)(1 + .07)(1 – .13)(1 – .04)(1 + .26)](1/6) – 1 = .0730,
or 7.30%
