An investment strategy has an expected return of 15 percent and a standard deviation of 12 percent. Assume investment returns are bell shaped.

Question

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.)

Answer 1

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.