Lessons From Coin-Tossers
Passive investors often use a thought experiment involving coin-tossers to make the point that you can't beat the market. It usually goes something like this. Imagine a stadium full of people who each hold a coin. Everyone is asked to toss their coin and those who throw tails are asked to leave. At the end of 10 tosses, there will be a number of people who tossed heads 10 times in a row. Then they ask, would you bet on these people to toss more than than the average number of heads in the next 10 coin-tosses? Their point is that just as you wouldn't bet on people who have a stellar track record of thowing heads to beat the average in the future. you should not bet on an investment manager, even one with a stellar investment track record, to beat the market.
It sounds persuasive until you realize that once you assume that everyone is tossing a normal coin you've already assumed that there is no skill involved because no one can be "better" at tossing such a coin than anyone else. Dr. Hersh Shefrin, a professor at Santa Clara University, uses a similar thought experiment in his book, Beyond Greed and Fear to reach a very different conclusion. Let me try to explain.
Let’s suppose we have a group of 42,000 people who each receive one of three types of coins - gold, silver, or bronze. The gold coins are weighted so they have a 60% chance of coming up heads. The silver coins are weighted evenly to give a 50% chance of tossing heads. The bronze coins are weighted to come up heads only 40% of the time.
If everyone tossed their coin 10 times, the group of 42,000 people would average 5 heads. If we want to bet on someone to beat the average of 5 heads out of the next 10 tosses, it would be smart to bet on someone holding a gold coin because each of these people can be expected to throw 6 heads. But, there is one more twist: all the coins are painted green so you can’t tell what kind of coin anyone has. Now, let's see if a track record has any value.
At the start, there are 14,000 people who have a gold coin, 14,000 with a silver coin, and 14,000 with a bronze coin. If you had to choose someone to bet on now, your chances of selecting someone with a gold coin are 33%. But if you wait until after the first coin toss the odds improve to 40%. Here's why. Of the 14,000 people holding gold coins, 60% will throw heads so they will comprise 8,400 of the people still in the stadium after the first toss. Of the 14,000 who hold a silver coin, 50% will throw heads, so 7,000 of them will remain. Of the bronze coin holders, 40% will throw heads accounting for 5,600 of those remaining. After the first toss, there will be 21,000 people in the stadium, but 8.400 of them, 40%, will be holding gold coins.
With each successive toss, the odds of selecting a gold coin holder from the people who remain improve. After the 10th coin toss, there will be 100 people in the stadium. But, 85 of them will have gold coins, 14 will have silver coins and 1 will have a bronze coin. If you select randomly from among the people who threw 10 heads in a row, your chances of picking someone with a gold coin holder are now 85%.
Restricting your choices to those with a track record of throwing 10 heads in a row greatly increases your odds of selecting a gold coin holder who can then be expected to throw 6 heads in the next 10 tosses and thus beat the average of all 42,000 which will be 5.
Since many who hold gold coins will not throw 6 heads in the next 10 tosses, even though they can be expected to, there is a good argument to choose all 100 of the people who threw 10 heads in a row. The expected performance of the entire group of 100 in the next 10 tosses is 5.84. By choosing all 100, the probability of the group averaging more than 5 is higher than if you just choose a single person with a gold coin.
I don't mean to imply that anyone should select managers based solely on their past performance. Investing is a lot harder than tossing coins. But if investing requires skill, then track records are a useful tool for finding those who are skilled.