Y Safle

You are in a game show, and at the end of the show you are allowed to choose one of three possible doors. Behind two of the doors are goats, and behind the third is a cash prize (which is what you want to win!). You choose a door, but instead of opening the door, the game show host will always open one of the remaining two doors, showing you a goat.

You now have two doors remaining, and the game show host offers you the chance to switch doors. The question is: is there any advantage to switching?

Yesterday I wrote about an error in Thomas Kida’s book over probability. Probability is a tricky thing, and the question above is a rephrasing of what has come to be called the “Monty Hall Problem” (after an American game show, which had a slightly different version of this problem on it).

This problem was quite famously presented by Marilyn Vos Savant in “Parade” magazine - a syndicated magazine found in many American newspapers. She presented the right answer to the problem above, and received a barrage of letters from some very clever people (including professors of Mathematics) arguing she was wrong. Eventually many of these professors wrote back to apologise for their mistake, but the experience shows that even the brightest minds can be fooled by the tricks of probability.

Because against all intuition, you should indeed switch.

Why?

There are three doors. The probability of the cash prize being behind the door you choose is 1/3. You choose door A, and there is a 1/3 chance that you have now won, and a 2/3 chance that you have chosen a goat.

The presenter shows your door C has a goat. What is the probability that door B has the cash prize?

Because the presenter must always reveal one goat or another, the scenario is still the same scenario. He has shown you a goat from one of the two remaining doors, but the chance that you have chosen the prize is still 1/3. Nothing has changed. The presenter would show you a goat if you had chosen the cash prize or not.

So against all instinct (especially for anyone who would not be fooled by the gambler’s fallacy), you should change to door B. Your chances of winning are now 2/3.

I bought Thomas Kida’s book, “Don’t believe everything you think (The 6 basic mistakes we make in thinking)”, in part because the title was similar to my occasional “Mistakes we Make in Thinking” series.

There is some good stuff in the book. But there is also a rather worrying example of exactly the kind of sloppy thinking Kida is supposed to be warning us against. He spends considerable space on the gambler’s fallacy, and then launches into a discussion of the unpredictability of the Stock Market, and how research has shown that there is no evidence that highly paid fund managers add any value to an investment fund, and that over the long haul, no funds significantly beat the index.

All this may well be right, but Kida’s error lies in what he does with these data:

Oftentimes investors move their money into a fund that has experienced good recent performance. However, statistcs tell us that we have regression to the mean. That is, if a fund is currently outperforming the market, its performance is likely to drop in the future to bring it back to average. And so, if we buy into a fund right after it has posted recent gains, we’re likely to be in for a fall. In effect, going after strong past performance often means we take money out of funds that are likely to rebound, and put it into funds that are ready to drop.

Kida has misunderstood regression towards the mean, and has committed an error known as the gambler’s fallacy (which he had already discussed in an earlier chapter).

Let us suppose that fund manager’s are indeed irrelevent, and that a fund has a 50/50 chance of underperforming or overperforming the market each year. If this assumption is indeed correct - and this is indeed Kida’s argument, then whether the fund will outperform or underperform the market this year is entirely unconnected with whether the fund outperformed or underperfomed the market last year.

If Kida is correct, then it makes no difference in the long run whether we leave the money where it is or move it (except for dealing charges incurred of course), because all funds will eventually do equally well.

If we buy into a fund right after it has posted gains then it is wrong to expect that we are in for a big fall. We are just as likely to do well (or badly) as if we buy into a fund that recently posted very poor gains.

But what is regression towards the mean then?

If we take the whole “population” of funds, and we measure all their respective gains each year, we come up with a mean (average) gain for all funds. Now, suppose we choose the 100 best performing funds and measure their gains - because these are the best perfroming funds, their mean gain will, of course, be higher than the mean for all funds.

Let us suppose that their mean gain was twice that of all funds.

Now next year we measure these means again. The mean gain for all funds and the mean for what were last year’s 100 best funds. What we find is that the mean for the 100 best funds of last year is now much closer to the mean of all funds. If fund performance is entirely random then that mean may be less than the mean for all funds, or more - but it will almost certainly be less than twice that of all funds.

Why does this happen? Because there was nothing special about the 100 best funds, and there is no guarantee that the funds that did well last year will do well this year. Thus their average should approach the population average.

But any individual fund could still be in the top 100 - and we would expect that to be the case. Regression towards the mean is only concerned with averages.

Still not convinced?

By Kida’s principle - moving money into an outperforming fund sets you up for a fall. Thus it would follow that moving money into an underperforming fund will set you up for a gain. Therefore, one should put money into the underperforming funds as the best strategy for success.

But it doesn’t work. Because Kida is wrong.

If the performance of a fund is random, the best strategy for success is to buy the fund with the lowest charges and leave your money where it is (or better still - just buy the shares that all the funds hold, and hold the shares).