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.

Question

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.)

Answer 1

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.