Exercise 3-65 Algo
The historical returns on a portfolio had an average return of 9 percent and a standard deviation of 13 percent. Assume that returns on this portfolio follow a bell-shaped distribution.
a. Approximately what percentage of returns were greater than 48 percent? (Round your answer to the nearest whole percent.)
b. Approximately what percentage of returns were below −4 percent? (Round your answer to 1 decimal place.)
a. To find the percentage of returns greater than 48 percent, we need to calculate the area under the bell curve to the right of 48 percent. Since we are given the average return and standard deviation, we can use the Z-score formula to calculate the Z-score for 48 percent.
Z = (x - μ) / σ
where x is the value (48 percent), μ is the mean (9 percent), and σ is the standard deviation (13 percent).
Z = (48 - 9) / 13 = 2.999
Using a Z-table or calculator, we can find that the area to the left of a Z-score of 2.999 is approximately 0.9987. To find the area to the right of 48 percent, we subtract this value from 1:
1 - 0.9987 = 0.0013
So, approximately 0.13 percent of returns were greater than 48 percent.
b. To find the percentage of returns below -4 percent, we need to calculate the area under the bell curve to the left of -4 percent. Again, we can use the Z-score formula to calculate the Z-score for -4 percent.
Z = (x - μ) / σ
where x is the value (-4 percent), μ is the mean (9 percent), and σ is the standard deviation (13 percent).
Z = (-4 - 9) / 13 = -1.3077
Using a Z-table or calculator, we can find that the area to the left of a Z-score of -1.3077 is approximately 0.0951. Therefore, approximately 9.5 percent of returns were below -4 percent.
To solve this exercise, we can use the Z-score formula to find the percentage of returns greater than a certain value or below a certain value. The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z: Z-score
X: Value of interest
μ: Mean (average return)
σ: Standard deviation
a. To find the percentage of returns greater than 48 percent:
Z = (48 - 9) / 13
Z = 39 / 13
Z ≈ 3
Looking up the Z-score of 3 in a standard normal distribution table, we find that the corresponding percentage is approximately 99.73%. However, since we are looking for returns greater than 48 percent, we need to calculate the percentage on the other side of the mean:
Percentage = 100% - 99.73%
Percentage ≈ 0.27% (rounded to the nearest whole percent)
Therefore, approximately 0.27% of the returns were greater than 48 percent.
b. To find the percentage of returns below -4 percent:
Z = (-4 - 9) / 13
Z = -13 / 13
Z = -1
Looking up the Z-score of -1 in a standard normal distribution table, we find that the corresponding percentage is approximately 15.87%.
Therefore, approximately 15.9% (rounded to 1 decimal place) of the returns were below -4 percent.