13.32 The stock market. The annual rate of return on stock indexes (which combine many individual stocks) is very roughly Normal. Since 1945, the Standard & Poor’s 500 index has had a mean yearly return of 12.5%, with a standard deviation of 17.8%. Take this Normal distribution to be the distribution of yearly returns over a long period.
(a) In what range do the middle 95% of all yearly returns lie?
(b) The market is down for the year if the return on the index is less than zero. In what proportion of years is the market down?
(c) In what proportion of years does the index gain 25% or more?