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 determine the probability that you will lose money by investing in the asset, given a normal distribution with an average return of 12.2% and a standard deviation of 27.8%, we need to find the probability that the return is less than 0% (losing money).

Here's how to do it step-by-step using the NORMDIST function in Excel:

  1. Identify the values for the NORMDIST function:

    • Mean (μ\mu) = 12.2%
    • Standard deviation (σ\sigma) = 27.8%
    • Value at which we want to calculate the probability (X) = 0% (since we want the probability of a loss).
  2. Use the NORMDIST function: The function in Excel is written as:

    excel
    =NORM.DIST(X, Mean, Standard_Deviation, TRUE)

    In this case, we would write:

    excel
    =NORM.DIST(0, 0.122, 0.278, TRUE)
  3. Interpret the output:

    • This function will output the cumulative probability up to 0%, which gives the probability of having a return of 0% or less.

After inputting these values, the function will return a probability. This probability represents the likelihood (as a percentage) that you will lose money in any given year.

Let me know if you’d like to see the exact Excel output or if you have questions on any specific step!


Here are five related questions to expand on this topic:

  1. How would the probability change if the standard deviation were smaller, say 15%?
  2. What is the probability of achieving a return greater than 20% in a given year?
  3. How would we use NORMDIST to find the probability of returns between 10% and 15%?
  4. If the average return increased to 15%, how would that impact the probability of a loss?
  5. Can we use this approach to determine the probability of returns exceeding 30%?

Tip: To convert the probability to a percentage in Excel, multiply the output by 100.

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Math Problem Analysis

Mathematical Concepts

Probability
Normal Distribution

Formulas

NORM.DIST function in Excel: =NORM.DIST(X, Mean, Standard_Deviation, TRUE)

Theorems

Properties of Normal Distribution
Empirical Rule

Suitable Grade Level

Undergraduate level