Question

Andrew plans to retire in 40 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. What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40 years will exceed 10%? What is the probability that the mean return will be less than 5%? Write a conclusion in the context of the problem.

Answer 1

Answer:

There are 42.7% chances that the return will be less than or equal to 5% and only 26% chance that the returns would be greater than 10%.  

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 8.7% = 0.087

Standard Deviation, σ = 20.2%  = 0.202

We are given that the distribution annual returns of stocks is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(return greater than 10%)

P(x > 0.10)

$P( x > 0.10) = P( z > \displaystyle\frac{0.10 - 0.087}{0.202}) = P(z > 0.0643)$

$= 1 - P(z \leq 0.643)$

Calculation the value from standard normal z table, we have,  

$P(x > 610) = 1 - 0.740 = 0.26 = 26\%$

b) P(return less than 5%)

$P(x \leq 0.05) = P(z \leq \displaystyle\frac{0.05-0.087}{0.202}) = P(z \leq -0.183)$

Calculating the value from the standard normal table we have,

$P( x < 0.05) =0.427 = 42.7\%$

Conclusion:

Hence, there are 42.7% chances that the return will be less than or equal to 5% and only 26% chance that the returns would be greater than 10%.