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:

  1. Identify the values:

    • Mean (μ\mu) = 12.2%
    • Standard Deviation (σ\sigma) = 27.8%
    • Target return (X) = 0% (since we want to calculate the probability of returns being less than 0%)
  2. 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.
  3. 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:

  1. How would the probability change if the standard deviation increased?
  2. What is the probability of achieving a return greater than 20% with the given mean and standard deviation?
  3. How do you interpret the probability result in the context of financial risk?
  4. How can we calculate the 95% confidence interval for this asset’s return?
  5. 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