The average return for large-cap domestic stock funds over three years was 14.4%. Assume the three-year returns were normally distributed across funds with a standard deviation of 4.4%.

a) Find the probability an individual large-cap domestic stock fund had a three-year return of 10% or less.

b) How big does the return have to be to put a domestic stock fund in the top 25% for the three-year period?

Answer :

Given :

The population mean $(\mu )=14.4$

The population standard deviation $(\sigma )=4.4$

Solution :

a) The probability an individual large-cap domestic stock fund had a three-year return of 10% or less :

$$\text{P}(x\le 10)=\text{P}(\frac{x–\mu }{\sigma }\le \frac{10–14.4}{4.4})$$ $$=\text{P}(z\le -1)$$ $$=0.1587$$

b) Let us assume that the return have to be $x_0$ % to put a domestic stock fund in the top 25% for the three-year period.

$$\therefore \text{P}(x>x_0)=0.25$$ $$\therefore 1–\text{P}(x>x_0)=1–0.25$$ $$\therefore \text{P}(x<x_0)=0.75$$ $$\therefore \text{P}(z<\frac{x_0–\mu }{\sigma })=0.75$$ $$\therefore \frac{x_0–\mu }{\sigma }=0.674$$ $$\therefore \frac{x_0–14.4}{4.4}=0.674$$ $$\therefore x_0–14.4=0.674\times 4.4$$ $$\therefore x_0\approx 17.37\mathrm{\%}$$