Stock market randomness An interview in an investment magazine (In the Vanguard, Autumn 2003) asked mathematician John Allen Paulos, “What common errors do investors make?” He answered, “People tend not to believe that markets move in random ways. Randomness is difficult to recognize. If you have people write down 100 Hs and Ts to simulate 100 flips of a coin, you will always be able to tell a sequence generated by a human from one generated by real coin flips. When humans make up the sequence, they don’t put in enough consecutive Hs and Ts, and they don’t make the lengths of those runs long enough or frequent enough. And that is one of the reasons people look at patterns in the stock market and ascribe significance to them.” (© The Vanguard Group, Inc., used with permission.)

a. Suppose that on each of the next 100 business days the stock market has a 1/2 chance of going up and a 1/2 chance of going down, and its behavior one day is independent of its behavior on another day. Use software, such as the web app mentioned in exercise 5.10 or random.org/coin to simulate whether the market goes up or goes down for each of the next 100 days.

What is the longest sequence of consecutive moves up or consecutive moves down that you observe?

b. Run the simulation nine more times, with 100 observations for each run, and each time record the longest sequence of consecutive moves up or consecutive moves down that you observe. For the 10 runs, summarize the proportion of times that the longest sequence was 1, 2, 3, 4, 5, 6, 7, 8, or more. (Your class may combine results to estimate this more precisely.)

c. Based on your findings, explain why if you are a serious investor you should not get too excited if sometime in the next few months you see the stock market go up for five days in a row or go down for five days in a row.