Question

The average return for large-cap domestic stock funds over the three years 2009-2011 was 14.3% . Assume the three-year returns were normally distributed across funds with a standard deviation of 4.5%.

a. What is the probability an individual large-cap domestic stock fund had a three-year return of at least 20% (to 4 decimals)?

b. What is the probability an individual large-cap domestic stock fund had a three-year return of 10% or less (to 4 decimals)?

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 2 decimals)?

Answer 1

a. What is the probability an individual large-cap domestic stock fund had a three-year return of at least 20% (to 4 decimals)?

In each case you need to convert the data to an appropriate Z-score. Z = ( value - mean ) / sdev

a) P(return > 20) = P(Z > ((20 - 14.3) / 4.5) = P(Z > 1.27)

Now look up 1.27 on the Normal Distribution table to find 0.898

P(Z > 1.27) = 1 - 0.8980 = 0.1020

b. What is the probability an individual large-cap domestic stock fund had a three-year return of 10% or less (to 4 decimals)?

b) Z = ( 10 - 14.3) / 4.5 = -0.96

Look up -0.96 on the table to find 0.1685

P(return < 10) = P(Z < -0.96) =0.1685

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 2 decimals)?

The return (in %) to put a domestic stock fund in the top 10% for the three-year period is the value of x when the area of the normal distribution at the right of x is equal to 0.1 (10%).

Then, we should find the value of x so that P(X>x) =P(Z>z) =10

P(X>x) = P(Z>z) =10.

Because P (Z > z) =1−P (Z ≤ z), then:

1−P (Z ≤ z) =0.1

⇒P (Z ≤ z) =1−0.1

⇒P (Z ≤ z) =0.9

⇒z=1.28

1.28 = ( value - mean ) / sdev = ( X - 14.3 ) / 4.5, then solve for X

X = ( 1.28 * 4.5 ) + 14.3 = 20.06%

A return of $20.06 percent or higher will put a domestic stock fund in the top 10 percent.