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CHAPTER 10
SOME LESSONS FROM CAPITAL
MARKET HISTORY
Answers to Concepts Review and Critical Thinking Questions
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.
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.
6. Before the fact, for most assets, the risk premium will be positive; investors demand compensation
7. Yes, the stock prices are currently the same. Below is a diagram that depicts the stocks’ price
(.9)(1.1) P0, while Standard Fixtures’ stock price increased by 10 percent, or (1.1)(.9) P0. Today,
each of the stocks is worth 99 percent of its original value.
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
9. To calculate an arithmetic return, you sum the returns and divide by the number of returns. As such,
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
2. The dividend yield is the dividend divided by the price at the beginning of the period, so:
Dividend yield = $1.95/$76
3. Using the equation for total return, we find:
R = [($68 – 76) + 1.95]/$76
R = –.0796, or –7.96%
And the dividend yield and capital gains yield are:
4. The total dollar return is the change in price plus the coupon payment, so:
Total dollar return = $1,052 – 1,010 + 49
Total dollar return = $91
5. The nominal return is the stated return, which is 12.1 percent. Using the Fisher equation, the real return
was:
6. Using the Fisher equation, the real returns for government and corporate bonds were:
(1 + R) = (1 + r)(1 + h)
7. The average return is the sum of the returns, divided by the number of returns. The average return for each
stock was:
1
.12 .24 .27 .14 .19 .0840, or 8.40%
5
N
i
i
X x N
=
+−++
= = =
We calculate the variance of each stock as:
( )
( )
2
2
1
1
N
Xi
i
x x N
=
= − −
The standard deviation is the square root of the variance, so the standard deviation of each stock is:
X = .041331/2
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
1976 23.93 5.07 18.86
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%
And the average return for T-bills over this period was:
b. Using the equation for variance, we find the variance for large company stocks over this period
was:
Variance = 1/5[(–.1469 – .0324)2 + (–.2647 – .0324)2 + (.3723 – .0324)2 + (.2393 – .0324)2 +
(–.0716 – .0324)2 + (.0657 – .0324)2]
Variance = .058136
And the standard deviation for T-bills over this period was:
c. The average observed risk premium over this period was:
Average observed risk premium = –19.90%/6
Average observed risk premium = –3.32%
9. a. To find the average return, we sum all the returns and divide by the number of returns, so:
Arithmetic average return = (.19 + .24 + .11 – .09 + .13)/5
Arithmetic average return = .1160, or 11.60%
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)
RP
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)
rp
12. Applying the five-year holding-period return formula to calculate the total return of the stock over the
five-year period, we find:
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 20 years to maturity. Using semiannual compounding, the
price today is:
14. The return of any asset is the increase in price, plus any dividends or cash flows, all divided by the
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 = ($91.45 – 82.18)/$82.18
R = .1128, or 11.28%
This is the return for three months, so the APR is:
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
17. Looking at the long-term corporate bond return history in Table 10.2, we see that the mean return was
6.4 percent, with a standard deviation of 8.3 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:
18. Looking at the large-company stock return history in Table 10.2, we see that the mean return was 12.1
percent, with a standard deviation of 19.8 percent. The range of returns you would expect to see 68
19. 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:
20. The arithmetic average return is the sum of the known returns divided by the number of returns, so:
Arithmetic average return = (.23 + .11 + .37 – .03 + .22 – .17)/6
Arithmetic average return = .1217, or 12.17%
21. To calculate the arithmetic and geometric average returns, we must first calculate the return for each
year. The return for each year is:
R1 = ($68.13 – 64.12 + 1.15)/$64.12 = .0805, or 8.05%
R2 = ($61.23 – 68.13 + 1.25)/$68.13 = –.0829, or –8.29%
22. To find the real return we need to use the Fisher equation. Re-writing the Fisher equation to solve for
the real return, we get:
r = [(1 + R)/(1 + h)] – 1
So, the real return each year was:
Year
T-bill return
Inflation
Real return
1973
.0729
.0871
–.0131
.0507
.0486
.0020
1977
.0545
.0670
–.0117
1978
.0764
.0902
–.0127
a. The average return for T-bills over this period was:
Average return = .6197/8
Average return = .0775, or 7.75%
b. Using the equation for variance, we find the variance for T-bills over this period was:
Variance = 1/7[(.0729 – .0775)2 + (.0799 – .0775)2 + (.0587 – .0775)2 + (.0507 – .0775)2 +
(.0545 – .0775)2 + (.0764 – .0775)2 + (.1056 – .0775)2 + (.1210 − .0775)2]
Variance = .000616
And the standard deviation for T-bills was:
c. The average observed real return over this period was:
Average observed real return = –.1120/8
Average observed real return = –.0140, or –1.40%
23. To find the return on the coupon bond, we first need to find the price of the bond today. Since one year
has elapsed, the bond now has six years to maturity, so the price today is:
P1 = $58(PVIFA5.4%,6) + $1,000/1.0546
P1 = $1,020.05
24. Looking at the long-term government bond return history in Table 10.2, we see that the mean return
was 6 percent, with a standard deviation of 9.9 percent. In the normal probability distribution,
approximately 2/3 of the observations are within one standard deviation of the mean. This means that
1/3 of the observations are outside one standard deviation away from the mean. Or:
Pr(R < –3.9 or R > 15.9) 1/3
But we are only interested in one tail here, that is, returns less than –3.9 percent, so:
Pr(R < –3.9) 1/6
25. The mean return for small company stocks was 16.5 percent, with a standard deviation of 31.7 percent.
Doubling your money is a 100% return, so if the return distribution is normal, we can use the z-statistic.
So:
z = (X – µ)/
26. It is impossible to lose more than 100 percent of your investment. Therefore, return distributions are
truncated on the lower tail at –100 percent.
27. Using the z-statistic, we find:
z = (X – µ)/
28. For each of the questions asked here, we need to use the z-statistic, which is:
z = (X – µ)/
a. z1 = (10% – 6.4%)/8.3% = .4337
This z-statistic gives us the probability that the return is less than 10 percent, but we are looking
for the probability the return is greater than 10 percent. Given that the total probability is 100
For a return less than 0 percent:
b. The probability that T-bill returns will be greater than 10 percent is:
z3 = (10% – 3.4%)/3.1% = 2.1290
c. The probability that the return on long-term corporate bonds will be less than –4.18 percent is:
z5 = (–4.18% – 6.4%)/8.3% = –1.2747
