Question

STATE: 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 from 1966 to 2015, the annual returns on S&P 500 had mean 11.0% and standard deviation 17.0% . PLAN: 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. We can use the Central Limit Theorem to make an inference. SOLVE: What is the probability, ?1 , assuming that the past pattern of variation continues, that the mean annual return on common stocks over the next 40 years will exceed 10% ? (Enter your answer rounded to two decimal places.) ?1= What is the probability, ?2 , that the mean return will be less than 5% ? (Enter your answer rounded to two decimal places.)

Answer 1

What is the probability, ?1 , assuming that the past pattern of variation continues, that the mean annual return on common stocks over the next 40 years will exceed 10% ?

Given data,

Mean = 11%, S.D = 17.0%

x = 10%

By applying normal distribution:-

z = - 0.372

P(z > - 0.372) = 0.645

The probability that the mean annual return on common stocks over the next 40 years will exceed 10% is 0.645.

What is the probability, ?2 , that the mean return will be less than 5% ?

Given data,

Mean = 11%, S.D = 17.0%

x = 5%

By applying normal distribution:-

z = - 2.23

P(z < - 2.23) = 0.013

The probability that the mean annual return on common stocks over the next 40 years will be less than 5% is 0.013.

There are about a 64% chance of getting average returns over 10% and a 1% chance of getting average returns less than 5%.