Math Problem Statement
Suppose the returns on a particular asset are normally distributed. The asset had an average return of 12.2 percent and a standard deviation of 27.8 percent. Use the NORMDIST function in Excel®raise to the power of ® to determine the probability that in any given year you will lose money by investing in this asset.
Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.
Solution
To find the probability that the return on this asset will be less than 0% (i.e., you’ll lose money), given a normally distributed return with a mean of 12.2% and a standard deviation of 27.8%, we can use the cumulative distribution function (CDF) of the normal distribution.
In Excel, we can use the NORM.DIST
function for this purpose. Here’s the approach:
-
Identify the values:
- Mean () = 12.2%
- Standard Deviation () = 27.8%
- Target return (X) = 0% (since we want to calculate the probability of returns being less than 0%)
-
Excel Formula Setup: Use the
NORM.DIST
function to calculate the probability:excel=NORM.DIST(0, 0.122, 0.278, TRUE)
- The first argument (0) is the value we want to find the cumulative probability for (i.e., losing money).
- The second argument (0.122) is the mean, in decimal form.
- The third argument (0.278) is the standard deviation, also in decimal form.
- The last argument (
TRUE
) specifies that we want the cumulative distribution.
-
Interpret the Result: After applying the function, Excel will give you a result in decimal form. Multiply by 100 to convert it to a percentage and round to two decimal places.
Answer:
The resulting value will be the probability, expressed as a percentage, that the asset will return less than 0% in a given year.
Would you like additional guidance on using this in Excel, or further details on how the normal distribution applies to investment risk?
Related Questions:
- How would the probability change if the standard deviation increased?
- What is the probability of achieving a return greater than 20% with the given mean and standard deviation?
- How do you interpret the probability result in the context of financial risk?
- How can we calculate the 95% confidence interval for this asset’s return?
- What does the shape of a normal distribution tell us about extreme positive or negative returns?
Tip:
In Excel, use NORM.INV
to find the value associated with a specific cumulative probability in a normal distribution.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Probability
Cumulative Distribution Function (CDF)
Formulas
NORM.DIST(x, mean, standard_deviation, cumulative)
Theorems
Normal Distribution Theorem
Suitable Grade Level
Grades 10-12
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