Math Problem Statement

An asset has an average return of 10.55 percent and a standard deviation of 20.46 percent. What range of returns should you expect to see with a 95 percent probability?

Multiple Choice

−20.14% to 41.24%

−30.37% to 51.47%

−9.91% to 11.19%

−50.83% to 71.93%

−9.91% to 31.01%

Solution

To find the range of returns with a 95 percent probability, we can use the concept of a confidence interval based on the mean and standard deviation. For a 95% confidence level, we generally use a range of ±1.96 standard deviations from the mean.

The formula for the range is:

Range=Mean±(1.96×Standard Deviation)\text{Range} = \text{Mean} \pm (1.96 \times \text{Standard Deviation})

Given:

  • Mean (average return) = 10.55%
  • Standard deviation = 20.46%

Calculating the Range

  1. Calculate 1.96×20.46%1.96 \times 20.46\%: 1.96×20.46%=40.1016%1.96 \times 20.46\% = 40.1016\%

  2. Apply this range to the mean:

    • Lower bound = 10.55%40.1016%=29.5516%10.55\% - 40.1016\% = -29.5516\%
    • Upper bound = 10.55%+40.1016%=50.6516%10.55\% + 40.1016\% = 50.6516\%

Thus, the expected range of returns with 95% probability is approximately: 29.55% to 50.65%-29.55\% \text{ to } 50.65\%

Closest Answer:

The closest answer to this range is: −30.37% to 51.47%

Would you like further explanation, or do you have any questions?


Here are five related questions to deepen your understanding:

  1. How would the range change if we calculated it for a 68% confidence level?
  2. What does a standard deviation of 20.46% imply about the asset's volatility?
  3. How would you interpret a 99% confidence interval for this return distribution?
  4. How do changes in the average return impact the confidence interval?
  5. Why is the 1.96 multiplier used for a 95% confidence interval?

Tip: The standard deviation reflects the variability around the mean. A higher standard deviation means a wider confidence interval, indicating greater risk in the returns.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Probability
Confidence Interval
Statistics

Formulas

Range = Mean ± (1.96 × Standard Deviation)

Theorems

68-95-99.7 Rule (Empirical Rule)

Suitable Grade Level

Grades 10-12