The average return for large-cap domestic stock funds over the three years 2009–2011 was
14.4% (AAII Journal, February, 2012). Assume the three-year returns were normally
distributed across funds with a standard deviation of 4.4%.
a. What is the probability an individual large-cap domestic stock fund had a three-year
return of at least 20%?
b. What is the probability an individual large-cap domestic stock fund had a three-year
return of 10% or less?
c. How big does the return have to be to put a domestic stock fund in the top 10% for the
three-year period?
To solve these questions, we will use the concept of the standard normal distribution, also known as the z-distribution. We will convert the given values into z-scores and then use statistical tables to find the probabilities.
a. To find the probability that an individual large-cap domestic stock fund had a three-year return of at least 20%, we need to calculate the z-score for 20%. The z-score formula is given by:
z = (x - μ) / σ
where x is the value we want to convert to a z-score, μ is the mean (average), and σ is the standard deviation.
Substituting the given values:
x = 20% (0.20)
μ = 14.4% (0.144)
σ = 4.4% (0.044)
z = (0.20 - 0.144) / 0.044
z ≈ 1.273
The next step is to find the probability corresponding to this z-score. You can use a standard normal distribution table or a calculator to find the area to the right of the z-score 1.273. This represents the probability of getting a value greater than 1.273.
b. To find the probability that an individual large-cap domestic stock fund had a three-year return of 10% or less, we need to calculate the z-score for 10%.
x = 10% (0.10)
z = (0.10 - 0.144) / 0.044
z ≈ -0.977
Using a standard normal distribution table or a calculator, you can find the area to the left of the z-score -0.977. This gives you the probability of getting a value less than or equal to -0.977.
c. To find the return that puts a domestic stock fund in the top 10% for the three-year period, we need to find the z-score that corresponds to the top 10% of the distribution.
The z-score corresponding to the top 10% can be determined using a standard normal distribution table or a calculator. This z-score will represent the value that separates the lowest 90% from the top 10% of the distribution.
Once you have the z-score, you can convert it back to the actual return using the z-score formula:
x = (z * σ) + μ
where x is the return, z is the z-score, σ is the standard deviation, and μ is the mean.
This value of x will be the return required to put a domestic stock fund in the top 10% for the three-year period.