Andrew plan to retire in 40 years. He is thinking of investing his retirement funds in stocks, so he seek out information on past returns. He learns that over the 101 years from 1900 to 2000, the real (that is, adjusted for inflation) returns on U.S. common stocks had mean 8.7% ans 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%?

1. i presume the standard deviation has been expressed
as % of the mean, else the problem is weird.
the SD has thus been taken as 1.7574 %

2. reference will have to be made to a z-table for normal df

for a return > 10%, z-value = [10 – 8.7 ] / 1.7574 = 0.74
looking up the right tail of the z-table,
probability = 0.2296, say 23 %

for a return < 5%, z value = [ 5 – 8.7 ] / 1.7574 = -2.105
looking up the left tail of the z-table ,
and interpolating between -2.10 & – 2.11,
probability = 0.0176 , say 1 3/4 %

Ans: (a) 23 % (b) 1 3/4 %