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:
Given:
- Mean (average return) = 10.55%
- Standard deviation = 20.46%
Calculating the Range
-
Calculate :
-
Apply this range to the mean:
- Lower bound =
- Upper bound =
Thus, the expected range of returns with 95% probability is approximately:
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:
- How would the range change if we calculated it for a 68% confidence level?
- What does a standard deviation of 20.46% imply about the asset's volatility?
- How would you interpret a 99% confidence interval for this return distribution?
- How do changes in the average return impact the confidence interval?
- 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.
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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
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