Question

Suppose that 1,000,000 people select which stocks to buy and hold for the year by throwing darts at the stock page. Suppose, too, that in any given year: The average stock price rises by 7 percent. In any given year, 50 percent of the dart throwers will have a return that is average or better. In any given year and by luck alone, 20 percent of the dart throwers will "beat the average" by 5 percentage points or more. a. After five years, how many people will report that they've earned 7 percent (the market average) or more on their stocks in every one of the previous five years? b. After five years, how many people will report that they've earned 12 percent \((5\) percent above the average) or more on their stocks in every one of the previous five years? c. After five years, how many people will report that they've earned 17 percent or more on their stocks in every one of the previous five years?

Short answer

a. Approximately 31,250 investors would report earning 7% or more on their stocks consistently over five years. b. Approximately 3.2 investors would report earning 12% or more on their stocks consistently over five years. c. Essentially, no investors would consistently earn 17% or more over five years, as the figure is insignificantly small.

Step-by-step solution

Calculate for 7 percent or more

Given that 50 percent of investors will earn the average return of 7 percent or more in any given year, we can calculate the number of investors who would achieve this consistently over five years by raising the probability to the power of the number of years. In other words, \(0.5^5\). After calculating that, it should be multiplied by the total number of investors, which is 1,000,000. So the formula would be \(1,000,000 * (0.5^5)\), which would yield the result.

Calculate for 12 percent or more

The exercise also states that 20 percent of investors will earn 5 percent above average, or 12 percent, in any given year. Again, calculate the number of investors who will achieve this over five years by raising this probability to the power of the number of years. In other words, \(0.2^5\). Multiply the result by 1,000,000 to get the number of investors. So the formula would be \(1,000,000 * (0.2^5)\), which would yield the result.

Calculate for 17 percent or more

Lastly, the exercise states that 10 percent of investors will beat the average by 10 percentage points, or earn 17 percent, in any given year. Using the same method as in the previous steps, calculate the number of investors who will achieve this consistently over five years by raising this probability to the power of the number of years, \(0.1^5\). Multiply the result by 1,000,000 to get the number of investors. So the formula would be \(1,000,000 * (0.1^5)\), which would yield the result.