Problem 1: Is this a Rare Event?

Dus and Phelps (2020 SBBI Yearbook) report that the year 1933 is missing in their histogram of returns for SMALL CAP STOCKS dating back to 1926. The reason given is that the annual return in 1933 was 142.9 percent, or rounded to 143 percent. Supposing small cap stocks have the sample mean or arithmetic average return 15.1 percent and standard deviation 22.4 percent, that the returns are independent draws from a normal distribution with those parameters (remember, the variance is the square of the standard deviation), answer the following question:

Is the observed returns value for 1933 a rare event? Put differently, do you expect to observe this event if the assumption that returns are random draws from the normal distribution with mean 15.1 percent and standard deviation 22.4 percent (compute its variance)?

You need a definition of rare event to answer this: Use the following: An observed annual return is a rare event if it exceeds the mean return plus or minus 3 times the standard deviation of returns. Observed rare events may be largeor unusualgains or losses according to the 3 -- criterion.

Briefy explain your answer.

Hint: Recall from our notes that for a N (0, 1) normal distribution almost 100 percent of the probability mass is concentrated in the range [-3:3] where you know = 1