Solutions Manual

Corporate Finance

Ross, Westerfield, and Jaffe

Asia Global Edition

01/30/2013

Prepared by:
Joe Smolira
Belmont University

1/30/2014

Revised by:

Joseph Lim
Singapore Management University

Ruth Tan
National University of Singapore
CHAPTER 10
RISK AND RETURN: LESSONS FROM
MARKET HISTORY
Answers to Concepts Review and Critical Thinking Questions

1. No, stocks are riskier. Some investors are highly risk averse, and the extra possible return doesn’t
attract them relative to the extra risk.

2. 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.

3. T-bill rates were highest in the early eighties. This was during a period of high inflation and is
consistent with the Fisher effect.

4. 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.

5. 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, P0. Over the first year, General
Materials’ stock price increased by 10 percent, or (1.1)  P0. Standard Fixtures’ stock price declined
by 10 percent, or (0.9)  P0. Over the second year, General Materials’ stock price decreased by 10
percent, or (0.9)(1.1)  P0, while Standard Fixtures’ stock price increased by 10 percent, or (1.1)(0.9)
 P0. Today, each of the stocks is worth 99 percent of its original value.

2 years ago 1 year ago Today

General Materials P0 (1.1)P0 (1.1)(0.9)P0 = (0.99)P0
Standard Fixtures P0 (0.9)P0 (0.9)(1.1)P0 = (0.99)P0

6. 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 10 -3

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.

7. 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.

8. 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 = [($86 – 75) + 1.20] / $75

R = .1627, or 16.27%

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

Dividend yield = $1.20 / $75

Dividend yield = .0160, or 1.60%

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

Capital gains yield = ($86 – 75) / $75

Capital gains yield = .1467, or 14.67%

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

R = [($67 – 75) + 1.20] / $75

R = –.0907, or –9.07%

And the dividend yield and capital gains yield are:

Dividend yield = $1.20 / $75

Dividend yield = .0160, or 1.60%
4 – SOLUTIONS MANUAL

Capital gains yield = ($67 – 75) / $75

Capital gains yield = –.1067, or –10.67%

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.

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

Total dollar return = $1,063 – 1,040 + 60

Total dollar return = $83

The total nominal percentage return of the bond is:

R = [($1,063 – 1,040) + 60] / $1,040

R = .0798, or 7.98%

Notice here that we could have simply used the total dollar return of $83 in the numerator of this
equation.

Using the Fisher equation, the real return was:

(1 + R) = (1 + r)(1 + h)

r = (1.0798 / 1.030) – 1
r = .0484, or 4.84%

5. The nominal return is the stated return, which is 11.80 percent. Using the Fisher equation, the real
return was:

(1 + R) = (1 + r)(1 + h)

r = (1.1180)/(1.031) – 1
r = .0844, or 8.44%

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

(1 + R) = (1 + r)(1 + h)

rG = 1.061/1.031 – 1
rG = .0291, or 2.91%

rC = 1.064/1.031 – 1
rC = .0320, or 3.20%
 CHAPTER 10 -5

7. The average return is the sum of the returns, divided by the number of returns. The average return for each
stock was:

N  .08  .21  .27  .11  .18  .0620, or 6.20%


X   xi  N 
 i 1  5

N  .12  .27  .32  .18  .24  .0980, or 9.80%


Y   yi  N 
 i 1  5

We calculate the variance of each stock as:

 N

 X 2     x i  x 2  N  1
 i 1 
X2 
1
5 1
 
.08  .062 2  .21  .062 2   .27  .062 2  .11  .062 2  .18  .062 2  .037170

Y 2 
1
5 1
 
.12  .098 2  .27  .098 2   .32  .098 2  .18  .098 2  .24  .098 2  .057920

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

X = (.037170)1/2
X = .1928, or 19.28%

Y = (.057920)1/2
Y = .2407, or 24.07%

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% 6.93% 21.62%
1974 –26.47 8.00 –34.47
1975 37.23 5.80 31.43
1976 23.93 5.08 18.85
1977 –7.16 5.12 –12.28
1978 6.57 7.18 –0.61
19.41% 38.11% –18.70%

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%
6 – SOLUTIONS MANUAL

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

T-bills average return = 38.11% / 6

T-bills average return = 6.35%

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 = 0.058136

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

Standard deviation = (0.058136)1/2

Standard deviation = 0.2411 or 24.11%

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

Variance = 1/5[(.0693 – .0635)2 + (.08 – .0635)2 + (.058 – .0635)2 + (.0508 – .0635)2 +

(.0512 – .0635)2 + (.0718 – .0635)2]
Variance = 0.000144

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

Standard deviation = (0.000144)1/2

Standard deviation = 0.012 or 1.2%

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

Average observed risk premium = –18.70% / 6

Average observed risk premium = –3.12%

The variance of the observed risk premium was:

Variance = 1/5[(–.2162 – (–.0312))2 + (–.3447 – (–.0312))2 + (.3143 – (–.0312))2 +

(.1885 – (–.0312))2 + (–.1228 – (–.0312))2 + (–.0061 – (–.0312))2]
Variance = 0.06183

And the standard deviation of the observed risk premium was:

Standard deviation = (0.06183)1/2

Standard deviation = 0.2487 or 24.87%

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

Arithmetic average return = (.27 +.13 + .18 – .14 + .09)/5

Arithmetic average return = .1060, or 10.60%
 CHAPTER 10 -7

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

Variance = 1/4[(.27 – .106)2 + (.13 – .106)2 + (.18 – .106)2 + (–.14 – .106)2 +

(.09 – .106)2]
Variance = 0.023430

So, the standard deviation is:

Standard deviation = (0.023430)1/2

Standard deviation = 0.1531, or 15.31%

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)

r = (1.1060/1.042) – 1
r = .0614, or 6.14%

b. The average risk premium is simply the average return of the asset, minus the average real risk-
free rate, so, the average risk premium for this asset would be:

RP  R – R f
RP = .1060 – .0510
RP = .0550, or 5.50%

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)

r f = (1.051/1.042) – 1
r f = .0086, or 0.86%

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:

rp  r – r f = 6.14% – 0.86%
rp = 5.28%

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 + R1)(1 + R2)(1 + R3)(1 + R4)(1 + R5)] – 1

5-year holding-period return = [(1 + .1612)(1 + .1211)(1 + .0583)(1 + .2614)(1 – .1319)] – 1
5-year holding-period return = 0.5086, or 50.86%
8 – SOLUTIONS MANUAL

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 24 years to maturity. Using semiannual compounding, the
price today is:

P1 = $1,000/1.04548
P1 = $120.90

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

R = ($120.90 – 109.83) / $109.83

R = .1008, or 10.08%

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 $4, so the return for the year was:

R = ($96.12 – 94.89 + 4.00) / $94.89

R = .0551, or 5.51%

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 = ($46.21 – 43.18) / $43.18

R = .0702, or 7.02%

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

APR = 4(7.02%)
APR = 28.07%

And the EAR is:

EAR = (1 + .0702)4 – 1
EAR = .3116, or 31.16%

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 0.0327 (0.0149) 0.0485
1927 0.0312 (0.0208) 0.0531
1928 0.0356 (0.0097) 0.0457
1929 0.0475 0.0020 0.0454
1930 0.0241 (0.0603) 0.0898
1931 0.0107 (0.0952) 0.1170
1932 0.0096 (0.1030) 0.1255
 CHAPTER 10 -9

So, the average real return was:

Average = (.0485 + .0531 + .0457 + .0454 + .0898 + .1170 + .1255) / 7

Average = .0750 or 7.5%

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 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:

R=  ± 1 = 6.4% ± 8.3% = –1.90% to 14.70%

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 = 6.4% ± 2(8.3%) = –10.20% to 23.0%

18. Looking at the large-company stock return history in Table 10.2, we see that the mean return was
11.8 percent, with a standard deviation of 20.2 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 = 11.8% ± 20.2% = –8.40% to 32.00%

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 = 11.8% ± 2(20.2%) = –28.60% to 52.20%

Intermediate

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:

5(.11) = .12 – .21 + .09 + .32 + R

R = .23 or 23%

The missing return has to be 23 percent. Now we can use the equation for the variance to find:

Variance = 1/4[(.12 – .11)2 + (–.21 – .11)2 + (.09 – .11)2 + (.32 – .11)2 + (.23 – .11)2]
Variance = 0.04035

And the standard deviation is:

Standard deviation = (0.04035)1/2

Standard deviation = 0.2009, or 20.09%
10 – SOLUTIONS MANUAL

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

Arithmetic average return = (.27 + .12 + .32 –.12 + .19 –.31) / 6

Arithmetic average return = .0783, or 7.83%

Using the equation for the geometric return, we find:

Geometric average return = [(1 + R1) × (1 + R2) × … × (1 + RT)]1/T – 1

Geometric average return = [(1 + .27)(1 + .12)(1 + .32)(1 – .12)(1 + .19)(1 – .31)](1/6) – 1
Geometric average return = .0522, or 5.22%

Remember, the geometric average return will always be less than the arithmetic average return if the
returns have any variation.

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 = ($64.83 – 61.18 + 0.72) / $61.18 = .0714, or 7.14%

R2 = ($72.18 – 64.83 + 0.78) / $64.83 = .1254, or 12.54%
R3 = ($63.12 – 72.18 + 0.86) / $72.18 = –.1136, or –11.36%
R4 = ($69.27 – 63.12 + 0.95)/ $63.12 = .1125, or 11.25%
R5 = ($76.93 – 69.27 + 1.08) / $69.27 = .1262, or 12.62%

The arithmetic average return was:

RA = (0.0714 + 0.1254 – 0.1136 + 0.1125 + 0.1262)/5

RA = 0.0644, or 6.44%

And the geometric average return was:

RG = [(1 + .0714)(1 + .1254)(1 – .1136)(1 + .1125)(1 + .1262)]1/5 – 1

RG = 0.0601, or 6.01%

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

 CHAPTER 10 -11

So, the real return each year was:

Year T-bill return Inflation Real return

1973 0.0693 0.0880 –0.0187
1974 0.0800 0.1220 –0.0420
1975 0.0580 0.0701 –0.0121
1976 0.0508 0.0481 0.0027
1977 0.0512 0.0677 –0.0165
1978 0.0718 0.0903 –0.0185
1979 0.1038 0.1331 –0.0293
1980 0.1124 0.1240 –0.0116
0.5973 0.7433 –0.1460

a. The average return for T-bills over this period was:

Average return = 0.5973 / 8

Average return = .0747, or 7.47%

And the average inflation rate was:

Average inflation = 0.7433 / 8

Average inflation = .0929, or 9.29%

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

Variance = 1/7[(.0693 – .0747)2 + (.0800 – .0747)2 + (.0580 – .0747)2 + (.0508 – .0747)2 +

(.0512 – .0747)2 + (.0718 – .0747)2 + (.1038 – .0747)2 + (.1124  .0747)2]
Variance = 0.000534

And the standard deviation for T-bills was:

Standard deviation = (0.000534)1/2

Standard deviation = 0.0231, or 2.31%

The variance of inflation over this period was:

Variance = 1/7[(.0871 – .0929)2 + (.1234 – .0929)2 + (.0694 – .0929)2 + (.0486 – .0929)2 +

(.0670 – .0929)2 + (.0902 – .0929)2 + (.1329 – .0929)2 + (.1252  .0929)2]
Variance = 0.000946

And the standard deviation of inflation was:

Standard deviation = (0.000946)1/2

Standard deviation = 0.0308, or 3.08%

c. The average observed real return over this period was:

Average observed real return = –.1320 / 8

Average observed real return = –.0165, or –1.65%
12 – SOLUTIONS MANUAL

d. The statement that T-bills have no risk refers to the fact that there is only an extremely small
chance of the government defaulting, so there is little default risk. Since T-bills are short term,
there is also very limited interest rate risk. However, as this example shows, there is inflation
risk, i.e. the purchasing power of the investment can actually decline over time even if the
investor is earning a positive return.

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 = $70(PVIFA5.5%,6) + $1,000/1.0556
P1 = $1,074.93

You received the coupon payments on the bond, so the nominal return was:

R = ($1,074.93 – 1,080.50 + 70) / $1,080.50

R = .0596, or 5.96%

And using the Fisher equation to find the real return, we get:

r = (1.0596 / 1.032) – 1
r = .0268, or 2.68%

24. Looking at the long-term government bond return history in Table 10.2, we see that the mean return
was 6.1 percent, with a standard deviation of 9.7 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 < –3.6 or R > 15.8)  1/3

But we are only interested in one tail here, that is, returns less than –3.7 percent, so:

Pr(R < –3.6)  1/6

You can use the z-statistic and the cumulative normal distribution table to find the answer as well.
Doing so, we find:

z = (X – µ) / 

z = (–3.6% – 6.1) / 9.7% = –1.00

Looking at the z-table, this gives a probability of 15.87%, or:

Pr(R < –3.3)  .1587, or 15.87%

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

95% level: R=  ± 2 = 6.1% ± 2(9.7%) = –13.30% to 25.50%

 CHAPTER 10 -13

The range of returns you would expect to see 99 percent of the time is the mean plus or minus 3
standard deviations, or:

99% level: R=  ± 3 = 6.1% ± 3(9.7%) = –23.00% to 35.20%

25. The mean return for small company stocks was 16.5 percent, with a standard deviation of 32.3
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) / 

z = (100% – 16.5%) / 32.3% = 2.585 standard deviations above the mean

This corresponds to a probability of  0.487%, or about once every 205 years. Tripling your money
would be:

z = (200% – 16.5%) / 32.3% = 5.681 standard deviations above the mean.

This corresponds to a probability of (much) less than 0.1%. The actual answer is .0000007768%, or
about once every 1.287 million years.

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.

Challenge

27. Using the z-statistic, we find:

z = (X – µ) / 

z = (0% – 11.8%) / 20.2% = –0.5842

Pr(R ≤ 0)  27.95%

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% = 0.4337

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
percent (or 1), the probability of a return greater than 10 percent is 1 minus the probability of a
return less than 10 percent. Using the cumulative normal distribution table, we get:

Pr(R ≥ 10%) = 1 – Pr(R ≤ 10%) = 33.23%

14 – SOLUTIONS MANUAL

For a return less than 0 percent:

z2 = (0% – 6.4%) / 8.3 = –0.7711

Pr(R < 10%) = 1 – Pr(R > 0%) = 22.03%

b. The probability that T-bill returns will be greater than 10 percent is:

z3 = (10% – 3.6%) / 3.1% = 2.0645

Pr(R ≥ 10%) = 1 – Pr(R ≤ 10%) = 1 – .9805  1.95%

And the probability that T-bill returns will be less than 0 percent is:

z4 = (0% – 3.6%) / 3.1% = –1.1613

Pr(R ≤ 0)  12.28%

c. The probability that the return on long-term corporate bonds will be less than –4.18 percent is:

z5 = (–4.18% – 6.4%) / 8.4% = –1.2595

Pr(R ≤ –4.18%)  10.39%

And the probability that T-bill returns will be greater than 10.56 percent is:

z6 = (10.56% – 3.6%) / 3.1% = 2.2452

Pr(R ≥ 10.56%) = 1 – Pr(R ≤ 10.56%) = 1 – .9876  1.24%