The historical returns on a balanced portfolio have had an average return of 4% and a standard deviation of 13%. Assume that returns on this portfolio follow a normal distribution. [You may find it useful to reference the z table.]
a. What percentage of returns were greater than 17%? (Round your answer to 2 decimal places.)
b. What percentage of returns were below −22%? (Round your answer to 2 decimal places.)
To solve this problem, we need to calculate the z-scores for 17% and -22%, and then use the z-table to find the corresponding percentages.
a. First, let's calculate the z-score for 17%:
z = (x - μ) / σ
z = (17 - 4) / 13
z = 13 / 13
z = 1
Looking up a z-score of 1 in the z-table, we find that the percentage of returns greater than 17% is approximately 0.1587, or 15.87% (rounded to 2 decimal places).
b. Next, let's calculate the z-score for -22%:
z = (x - μ) / σ
z = (-22 - 4) / 13
z = -26 / 13
z = -2
Looking up a z-score of -2 in the z-table, we find that the percentage of returns below -22% is approximately 0.0228, or 2.28% (rounded to 2 decimal places).
To find the percentage of returns that were greater than 17%, we need to calculate the z-score for 17% and use the z-table to determine the corresponding percentage.
To calculate the z-score:
z = (x - μ) / σ
where:
x = value we want to find the percentage for (17% in this case)
μ = mean of the returns (4% in this case)
σ = standard deviation of the returns (13% in this case)
So,
z = (17 - 4) / 13 = 1
Next, we will use the z-table to find the percentage corresponding to a z-score of 1. From the z-table, a z-score of 1 corresponds to a percentage of approximately 0.8413.
Therefore, the percentage of returns greater than 17% is 0.8413 or 84.13%.
To find the percentage of returns that were below -22%, we follow the same process.
z = (-22 - 4) / 13 = -2
From the z-table, a z-score of -2 corresponds to a percentage of approximately 0.0228.
Therefore, the percentage of returns below -22% is 0.0228 or 2.28%.
To find the percentage of returns that were greater than 17%, we need to calculate the z-score for 17% and then use the z-table to find the corresponding percentage.
Step 1: Calculate the z-score:
The z-score is calculated as:
z = (x - μ) / σ
where x is the value we want to convert to a z-score, μ is the mean (average return), and σ is the standard deviation.
In this case, x = 17%, μ = 4%, and σ = 13%. Plugging these values into the formula, we get:
z = (17% - 4%) / 13%
Step 2: Use the z-table:
Using the z-table, we find the corresponding area under the standard normal distribution curve for the calculated z-score. The z-table provides the area to the left of the z-score. However, we want the area to the right of the z-score (since we are looking for returns greater than 17%). So, we subtract the area from 1 to get the area to the right.
Let's calculate it:
z = (17% - 4%) / 13% = 1.00
From the z-table, the area to the left of 1.00 is 0.8413. Therefore, the area to the right is:
Area to the right = 1 - 0.8413 = 0.1587
Step 3: Convert the area to a percentage:
To convert the area to a percentage, we multiply by 100:
Percentage = Area * 100 = 0.1587 * 100 = 15.87%
So, approximately 15.87% of returns were greater than 17%.
Now let's move on to part b.
To find the percentage of returns that were below -22%, we follow the same steps as above.
Step 1: Calculate the z-score:
We have x = -22%, μ = 4%, and σ = 13%. Plugging these values into the formula, we get:
z = (-22% - 4%) / 13%
Step 2: Use the z-table:
From the z-table, we find the area to the left of the calculated z-score.
Let's calculate it:
z = (-22% - 4%) / 13% = -2.00
From the z-table, the area to the left of -2.00 is 0.0228.
Step 3: Convert the area to a percentage:
To convert the area to a percentage, we multiply by 100:
Percentage = Area * 100 = 0.0228 * 100 = 2.28%
So, approximately 2.28% of returns were below -22%.