IAn investment strategy has an expected return of 9 percent and a standard deviation of 5 percent. Assume investment returns are bell shaped.
a. How likely is it to earn a return between 4 percent and 14 percent? (Enter your response as decimal values (not percentages) rounded to 2 decimal places.)
b. How likely is it to earn a return greater than 14 percent? (Enter your response as decimal values (not percentages) rounded to 2 decimal places.)
c. How likely is it to earn a return below −1 percent? (Enter your response as decimal values (not percentages) rounded to 3 decimal places.)
a. To find the likelihood of earning a return between 4 percent and 14 percent, we need to find the area under the bell curve between these two points. This can be calculated using the standard normal distribution.
First, we need to convert the values of 4 percent and 14 percent to standard deviations from the mean. The formula for this is:
Z = (X - μ) / σ
Where Z is the number of standard deviations from the mean, X is the value we want to convert, μ is the mean (expected return), and σ is the standard deviation.
For 4 percent:
Z1 = (4 - 9) / 5 = -1
For 14 percent:
Z2 = (14 - 9) / 5 = 1
Next, we need to find the area under the standard normal distribution curve between Z1 and Z2. We can use a standard normal distribution table, or a calculator with a built-in function for this.
For Z1 = -1, the area under the curve to the left of Z1 is 0.1587.
For Z2 = 1, the area under the curve to the left of Z2 is 0.8413.
Therefore, the area between Z1 and Z2 is: 0.8413 - 0.1587 = 0.6826.
So, the likelihood of earning a return between 4 percent and 14 percent is 0.6826, or 68.26%.
b. To find the likelihood of earning a return greater than 14 percent, we need to find the area under the curve to the right of Z2.
Using the standard normal distribution table, the area to the left of Z2 is 0.8413.
Therefore, the area to the right of Z2 is 1 - 0.8413 = 0.1587.
So, the likelihood of earning a return greater than 14 percent is 0.1587, or 15.87%.
c. To find the likelihood of earning a return below -1 percent, we need to find the area under the curve to the left of Z1.
Using the standard normal distribution table, the area to the left of Z1 is 0.1587.
So, the likelihood of earning a return below -1 percent is 0.1587, or 15.87%.
To compute the likelihood of earning a return within a specific range or below a certain percentage, we need to use the standard normal distribution table (also known as the Z-table or standard score table). Here's how to calculate the probabilities step by step:
a. To find the likelihood of earning a return between 4 percent and 14 percent, we need to calculate the probability of the returns falling within one standard deviation of the mean.
1. Calculate the Z-scores for both the lower and upper limits of the range using the formula: Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For the lower limit (4 percent):
Z1 = (4 - 9) / 5 = -1.
For the upper limit (14 percent):
Z2 = (14 - 9) / 5 = 1.
2. Look up the corresponding probabilities in the Z-table for both Z1 and Z2.
By referencing the Z-table, we find that the area to the left of Z = -1 is approximately 0.1587, and the area to the left of Z = 1 is approximately 0.8413.
3. Subtract the lower probability from the higher probability to find the probability within the range.
Probability = 0.8413 - 0.1587 = 0.6826.
Therefore, the likelihood of earning a return between 4 percent and 14 percent is 0.6826 (rounded to 2 decimal places).
b. To find the likelihood of earning a return greater than 14 percent, we need to calculate the probability beyond one standard deviation above the mean.
1. Calculate the Z-score for the upper limit (14 percent) as done before (Z2 = 1).
2. Subtract the area to the left of Z = 1 from 1 (total area under the curve) to find the probability.
Probability = 1 - 0.8413 = 0.1587.
Therefore, the likelihood of earning a return greater than 14 percent is 0.1587 (rounded to 2 decimal places).
c. To find the likelihood of earning a return below -1 percent, we need to calculate the probability beyond one standard deviation below the mean.
1. Calculate the Z-score for the lower limit (-1 percent) as done before (Z1 = -1).
2. Use the Z-table to find the area to the left of Z = -1, which is 0.1587.
Therefore, the likelihood of earning a return below -1 percent is 0.1587 (rounded to 3 decimal places).