An investment strategy has an expected return of 15 percent and a standard deviation of 12 percent. Assume investment returns are bell shaped.
a. How likely is it to earn a return between 3 percent and 27 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 27 percent? (Enter your response as decimal values (not percentages) rounded to 2 decimal places.)
c. How likely is it to earn a return below −9 percent? (Enter your response as decimal values (not percentages) rounded to 3 decimal places.)
To answer these questions, we can use the z-score formula.
a. To find the likelihood of earning a return between 3% and 27%, we need to calculate the z-score for both values and find the area under the normal distribution curve between these two z-scores. The z-score formula is (x - μ) / σ, where x is the value we want to find the probability for, μ is the mean/expected return (15% in this case), and σ is the standard deviation (12% in this case).
For 3%: z = (3 - 15) / 12 = -1
For 27%: z = (27 - 15) / 12 = 1
Now, we need to find the probability (area under the curve) between these two z-scores. We can use a standard normal distribution table or a calculator to look up these values.
Using a standard normal distribution table, the probability of earning a return between -1 and 1 (z-scores) is 0.6826.
So, the probability of earning a return between 3% and 27% is 0.6826.
b. To find the likelihood of earning a return greater than 27%, we need to calculate the z-score for 27% and find the area under the normal distribution curve to the right of this z-score.
For 27%: z = (27 - 15) / 12 = 1
Using a standard normal distribution table or a calculator, the probability of earning a return greater than 27% (to the right of the z-score 1) is 0.1587.
So, the probability of earning a return greater than 27% is 0.1587.
c. To find the likelihood of earning a return below -9%, we need to calculate the z-score for -9% and find the area under the normal distribution curve to the left of this z-score.
For -9%: z = (-9 - 15) / 12 = -2
Using a standard normal distribution table or a calculator, the probability of earning a return below -9% (to the left of the z-score -2) is 0.0228.
So, the probability of earning a return below -9% is 0.023.
To solve these questions, we can use the standard normal distribution with a mean of 0 and a standard deviation of 1.
a. First, we need to standardize the values of 3 percent and 27 percent:
Z1 = (3 - 15) / 12 = -1.00
Z2 = (27 - 15) / 12 = +1.00
Next, we can look up the corresponding probabilities in the standard normal distribution table:
P(3% < x < 27%) = P(-1.00 < Z < 1.00)
Using the standard normal distribution table, we find that the cumulative probability for Z = 1.00 is 0.8413, and the cumulative probability for Z = -1.00 is 0.1587. So,
P(-1.00 < Z < 1.00) = 0.8413 - 0.1587 = 0.6826
Therefore, the likelihood of earning a return between 3 percent and 27 percent is 0.6826 or 68.26% (rounded to 2 decimal places).
b. To find the probability of earning a return greater than 27 percent, we calculate the cumulative probability for Z > 1.00:
P(Z > 1.00) = 1 - P(Z < 1.00)
Using the standard normal distribution table, we find that the cumulative probability for Z = 1.00 is 0.8413. So,
P(Z > 1.00) = 1 - 0.8413 = 0.1587
Therefore, the likelihood of earning a return greater than 27 percent is 0.1587 or 15.87% (rounded to 2 decimal places).
c. To find the probability of earning a return below -9 percent, we calculate the cumulative probability for Z < -1.00:
P(Z < -1.00)
Using the standard normal distribution table, we find that the cumulative probability for Z = -1.00 is 0.1587.
Therefore, the likelihood of earning a return below -9 percent is 0.1587 or 15.87% (rounded to 3 decimal places).