Question 1

A stock had annual returns of 9 percent,−16 percent,14 percent,and 2 percent over the past 4 years.What is the standard deviation of these returns?

Accepted Answer

Mean = (0.09 − 0.16 + 0.14 + 0.02)/ 4 = .0225 σ = {[(0.09 − 0.0225)² + (−0.16 − 0.0225)² + (0.14 − 0.0225)² + (0.02 − 0.0225)²] / (4 − 1)}^.5 = 0.1312,or 13.12%

Question 2

Suppose you own a risky asset with an expected return of 18.2 percent and a standard deviation of 27.2 percent.If the returns are normally distributed,the approximate probability of losing 9 percent or more in a single year is ________ percent.

Accepted Answer

Lower point of 68.26% probability range = 18.2% − 27.2% = −9% Approximate probability of losing 15 percent or more = (100% − 68.26%)/ 2 = 15.87%

Question 3

For the period 1926 to 2015,small-company stocks had an average return of 16.5 percent,U.S.Treasury bills returned 3.5 percent,and inflation averaged 3.0 percent.What was the risk premium on small-company stocks during this period?

Accepted Answer

Risk premium = 16.5% − 3.5% = 13.0%

Question 4

A stock had returns of 8 percent,13 percent,and −3 percent for the past 3 years.Based on these returns,what is the approximate probability that this stock will earn at least 14.19 percent in any one given year?

Accepted Answer

Average return = (0.08 + 0.13 − 0.03)/ 3 = 0.06 σ = {[(0.08 − 0.06)² + (0.13 − 0.06)² + (−0.03 − 0.06)²] / (3 − 1)}^.5 = 0.0819 Upper end of the 68.26% probability range = 0.06 + 0.0819= 0.1419,or 14.19% Approximate probability of earning at least 14.19% = (100% − 68.26%)/ 2 = 15.87%

Question 5

Assume a $1 investment in a stock 23 years ago is now worth $28.82.What is the geometric average return for the period?

Accepted Answer

Geometric average return = 28.82^1/23 − 1 = 0.1574,or 15.74%

Question 6

A stock has an expected rate of return of 13.9 percent and a standard deviation of 23.3 percent.Which one of the following best describes the probability that this stock will lose more than 1/3 of its value in any one year?

Accepted Answer

Lower bound of 95.44 percent probability range = 0.139 − (2 × 0.233)= −0.327,or −32.7% Probability = (100% − 95.44%)/ 2 = 2.28% A loss of 1/3,or 33.3 percent,would be a tad further out into the tail than a loss of 32.7 percent,so a probability of 2.25 percent would be a reasonable approximation.

Question 7

What are the arithmetic and geometric average returns for a stock with annual returns of 6.2 percent,12.8 percent,−7.7 percent,and 9.4 percent?

Accepted Answer

Arithmetic average = (0.062 + 0.128 − 0.077 + 0.094)/ 4 = 0.05175,or 5.175% Geometric return = (1.062 × 1.128 × 0.923 × 1.094)^.25 − 1 = 0.04873,or 4.873%

Question 8

A stock returned 11 percent,16 percent,−7 percent,and −13 percent annually for the past 4 years.Based on this information,what is the 99.74 percent probability range for any one given year?

Accepted Answer

Mean = (0.11 + 0.16 − 0.07 − 0.13)/ 4 = 0.0175,or 1.75% σ ={[(0.11 − 0.0175)² + (0.16 − 0.0175)² + ( −0.07 − 0.0175)² + (−0.13 − 0.0175)²] / (4 − 1)}^.5 = 0.1394,or 13.94% 99.74% probability range = 1.75% ± (3 × 13.94%)= −40.06 to 43.56%

Question 9

Assume that during a 7-year period,inflation averaged 2.8 percent,U.S.Treasury bill yields averaged 3.3 percent,and long-term government bond yields averaged 5.9 percent.What was the average rate of return on long-term corporate bonds if they commanded a risk premium of 0.2 percent more than the long-term government bonds?

Accepted Answer

Long-term corporate bond yield = 5.9% + 0.2% = 6.1%

Question 10

What are the arithmetic and geometric average returns for a stock with annual returns of 14.9 percent,14.6 percent,−2.8 percent,and 9.7 percent?

Accepted Answer

Arithmetic average = (0.149 + 0.146 − 0.028 + 0.097)/ 4 = 0.091,or 9.1% Geometric return = (1.149 × 1.146 × 0.972 × 1.097)^.25 − 1 = 0.08854,or 8.854%

Question 11

Over the past 3 years,a stock had annual returns of 6.4 percent,11.9 percent,and 14.8 percent.What is the mean return?

Accepted Answer

Mean = (6.4% + 11.9% + 14.8%)/ 3 = 11.03%

Question 12

A stock had returns of 22 percent,1 percent,−27 percent,19 percent,and 7 percent annually for the past 5 years.Based on these returns,what is the approximate probability that this stock will earn at least 60 percent in any one given year?

Accepted Answer

Mean = (0.22 + 0.01 − 0.27 + 0.19 + 0.07)/ 5 = 0.044,or 4.4% Σ = {[(0.22 − 0.044)² + (0.01 − 0.044)² + (−0.27 − 0.044)² + (0.19 − 0.044)² + (0.07 − 0.044)²] / (5 - 1)}^.5 = 0.1954,or 19.54% Upper end of the 99.74% probability range = 4.4% + 3(19.54%)= 63.02% Probability of earning at least 63.02% = (100% − 99.74%)/ 2 = .13% The probability of earning 60 percent or more will be greater than the probability of earning 63.02 percent or more,so an approximation of 0.13 percent is reasonable.

Question 13

A stock had returns of 13 percent,8 percent,1 percent,and −9 percent annually for the past 4 years.What is the mean and standard deviation of these returns?

Accepted Answer

Mean = (0.13 + 0.08 + 0.01 − 0.09)/ 4 = 0.0325,or 3.25% σ= {[(0.13 − 0.0325)² + (0.08 − 0.0325)² + (0.01 − 0.0325)² + (−0.09 − 0.0325)²] / (4 − 1)}^.5 = 0.0954 or 9.54%

Question 14

The arithmetic average return on your portfolio for the past 5 years is 9.3 percent.You earned −7.4 percent,17.1 percent,6.3 percent,and 11.8 percent for 4 of those 5 years.What rate of return did you earn in the fifth year?

Accepted Answer

0.093= (−0.074 + 0.171 + 0.063 + 0.118+ R)/ 5 R = 0.187,or 18.7%

Question 15

Jen invested $1 fifty years ago.Today,her investment is worth $18,329.46.What is the geometric average return on this investment?

Accepted Answer

Geometric return = 18,329.46^1/50 − 1 = 0.2169,or 21.69%

Question 16

GLM stock provided annual returns of 9.9 percent,8.2 percent,−6.7 percent,4.5 percent,and 7.1 percent over the last 5 years.What is the geometric average rate of return?

Accepted Answer

Geometric average return = (1.099 × 1.082 × 0.933 × 1.045 × 1.071)^1/5 − 1 = 0.442,or 4.42%

Question 17

A stock had returns of 7 percent,7 percent,7 percent,and 7 percent annually over the past 4 years.What is the geometric average return for this time period?

Accepted Answer

The geometric average equals the arithmetic average when there is no deviation of returns.

Question 18

Assume inflation averaged 1.2 percent during a period in which U.S.Treasury bills earned 1.3 percent.What was the rate of return on large-company stocks if the risk premium on those stocks was 6.8 percent for the period?

Accepted Answer

R = 1.3% + 6.8% = 8.1%

Question 19

A stock had returns of 11 percent,26 percent,8 percent,and −38 percent annually for the past 4 years.What is the mean and variance of these returns?

Accepted Answer

Mean = (0.11 + 0.26 + 0.08 − 0.38)/ 4 = 0.0175,or 1.75% σ² = [(0.11 − 0.0175)² + (0.26 − 0.0175)² + (0.08 − 0.0175)² + (-0.38 − 0.0175)²] / (4 − 1)= 0.076425

Question 20

Assume a 20-year period produced small-company returns of 16.7 percent with a risk premium of 9.8 percent,inflation of 5.4 percent,and long-term government bond returns of 10.1 percent.What was the risk-free rate of return during this period?

Accepted Answer

R_f = 16.7% − 9.8% = 6.9%

Quiz 10: Risk and Return: Lessons From Market History