# Bayesian statistics: is the market a fair coin?

The current levels of market volatility and low expected return are problematic for everyone – investors, banks, governments. I want to emphasize those two things in combination are bad, its not just “volatility”, it’s the level of volatility, and the fact that investors aren’t sure they’ll be compensated for it over the foreseeable future.

The New York Times book review on Bayes and his theorem that dovetails well with this problem. Traditional (or “frequentist”) statistics usually starts with an example of a “fair” coin – one that has an objective 50/50 chance of heads or tails. They then run through a number of (sometimes counter-intuitive) statistical facts.

A common one is “The coin has just landed heads four times in a row. Is the coin more, less, or equally likely to land on tails next time?”. People often say “more likely to land on tails”, because they think that in order to be fair the coin must revert to a 50/50 balance in  a short period of time. The correct answer is that the coin has no memory- it is equally likely to land on both.

However, this problem is almost exactly the opposite of the problem which we actually face in the world. In the real world we don’t know the coin is fair, and we want to figure out what the odds and pay-offs are. When investing, we often use historic returns to estimate an expected return figure for the future – assuming that the future will be like the past.

But average returns in the stock market are not objectively known – we don’t know that the coin is fair. And in fact, the coin might be fair sometimes, and unfair other times. We don’t know when it changes either. So the fear facing some investors these days is that the market, has stopped being a positive expected return gamble. And if that is true, why play the game?

If someone else has said it perfectly, just quote them. In this case, Felix Salmon nails it.

Obviously, the stock market is a dangerous place for short-term speculation — and if you can’t afford to see a 5% drop in one day, or a 20% drop over the course of a few weeks, then you shouldn’t be investing in stocks at all. It’s not a place for money you’re likely to need to spend any time soon. But if you’re a long-term investor, the one advantage you have over the big institutions is that you don’t mark to market, and are therefore less likely to be forced to puke up liquid and valuable stocks when markets fall. Take advantage of that, stay calm when markets get volatile, and over the long term you’ll be glad.

So what is the long term? Probably at least 5 years, though 20 is even better.