The historical returns on a portfolio had an average return of 11 percent and a standard deviation of 18 percent. Assume that returns on this portfolio follow a bell-shaped distribution.
a. Approximately what percentage of returns were greater than 29 percent? (Round your answer to the nearest whole percent.)
b. Approximately what percentage of returns were below −25 percent? (Round your answer to 1 decimal place.)
a. To find the percentage of returns greater than 29 percent, we need to calculate the z-score for a return of 29 percent using the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the return value
- μ is the average return
- σ is the standard deviation
In this case, x = 29 percent, μ = 11 percent, and σ = 18 percent.
z = (29 - 11) / 18 ≈ 1
Now we need to find the area under the bell-shaped curve to the right of the z-score of 1. Since the distribution is symmetric, we can look up the area to the left of z = 1 in the standard normal distribution table to find the area to the right of z = 1.
The standard normal distribution table gives the area to the left of z = 1 as 0.8413. Therefore, the area to the right of z = 1 is 1 - 0.8413 = 0.1587.
To find the percentage of returns greater than 29 percent, we multiply the area by 100:
Percentage of returns greater than 29 percent = 0.1587 * 100 ≈ 15.9 percent
Therefore, approximately 15.9 percent of the returns were greater than 29 percent.
b. Similarly, to find the percentage of returns below -25 percent, we need to calculate the z-score for a return of -25 percent using the same formula:
z = (x - μ) / σ
In this case, x = -25 percent, μ = 11 percent, and σ = 18 percent.
z = (-25 - 11) / 18 ≈ -2
Now we need to find the area under the bell-shaped curve to the left of the z-score of -2. Looking up -2 in the standard normal distribution table gives us an area of 0.0228.
To find the percentage of returns below -25 percent, we multiply the area by 100:
Percentage of returns below -25 percent = 0.0228 * 100 ≈ 2.3 percent
Therefore, approximately 2.3 percent of the returns were below -25 percent.
Step 1: Calculate the Z-score
To find the percentage of returns greater than a given value, we first need to calculate the corresponding Z-score. The Z-score is a measure of how many standard deviations a value is from the mean.
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we are interested in
μ is the average return
σ is the standard deviation
For part a, we want to find the Z-score for X = 29. Applying the formula, we get:
Z = (29 - 11) / 18
Z = 18 / 18
Z = 1
Step 2: Find the percentage using the Z-table
The Z-table provides the percentage of values that fall below a certain Z-score. To find the percentage of returns greater than 29%, we need to look up the percentage corresponding to a Z-score of 1.
Using the Z-table, we find that the percentage corresponding to a Z-score of 1 is approximately 84.13%.
So, approximately 84% of returns were greater than 29%.
For part b, we follow the same steps but with X = -25.
Z = (-25 - 11) / 18
Z = -36 / 18
Z = -2
Using the Z-table, we find that the percentage corresponding to a Z-score of -2 is approximately 2.28%.
So, approximately 2.3% of returns were below -25%.