Question

Andrew plans to retire in 32 years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that over the entire 20th century, the real (that is, adjusted for inflation) annual returns on U.S. common stocks had mean 8.7% and standard deviation 20.2%. The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal. (a) What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 32 years will exceed 14%

Answer 1

Answer: 0.3974

Explanation:

Given : The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal.

Real annual returns on U.S. common stocks had mean :
`\mu=0.087`

Standard deviation :
`\sigma=0.202`

We assume that the past pattern of variation continues.

Let x be the random variable that represents the annual returns on common stocks over the next 32 years .

The formula for z-score :
`z=(x-\mu)/(\sigma)`

For x= 0.14,
`z=(0.14-0.087)/(0.202)\approx0.26`

By using the standard normal distribution table , we have

The probability that the mean annual return on common stocks over the next 32 years will exceed 14% :-

`P(x>0.14)=P(z>0.26)=1-P(z\leq0.26)\\\\=1-0.6025681=0.3974319\approx0.3974`