# The Magic of Mean Reversion: More Than Meets The Eye

April 29, 2014 Leave a comment

I flipped three heads in a row on the crowded floor at a conference. No big deal. The next toss comes up…heads. Some eyebrows are raised as I prepare to flip again…heads. That’s 5 in a row. A small crowd gathers. “He’s due for a tail” someone says as my thumb flicks the coin into the air…heads. “What are the odds?!” I hear. Someone chimes in – “the odds are growing that he’ll get tails next time.” The coin flips through the air once more…

The topic of mean-reverting investment strategies came up. These are strategies that rely on the market’s *eventual* return to expected/average (or ‘mean’) returns. If markets are really hot right now, the idea goes, we should expect them to cool down. If they’re cool, we should expect them to heat up. Conventional wisdom is that this happens naturally because the markets have to ‘go back to average.’ The tricky part, for me at least, is that while mean reversion may work in the aggregate, it doesn’t actually predict any one stock’s near-term returns at all.

How can something be random at the micro-level (a specific stock’s returns) but exhibit non-random behavior at the macro level (a market’s return to ‘normality’)? Let’s imagine that a given stock’s returns are really random: like flipping a fair coin. They are independent of one another and each trade has a 50/50 chance of being up or down (I know that’s an incorrect assumption – but it’s an instructive model to demonstrate the point). We all know that no matter how many times I flip the coin, the next flip has exactly a 50/50 chance of being heads (or tails). But the more heads I flipped in a row at that conference, the more everybody expected the next toss to come up tails. After all, the ‘average’ has to come back to 50/50. But that’s not how it works. The individual flips, like the individual returns of the individual stocks (in our simplistic model), are independent and each one has the exact same chance as every other flip – 50/50. The fact that I’ve just gotten five or six or any number of heads in a row does not change the next chance. There is no “he’s due for a tails” effect. There is no “the odds are growing that he’ll get a tail next time.” There is *no* reversion to the mean for the individual coin’s next flip.

But there *is* reversion to the mean *in the aggregate*. With 100 tosses, the odds are very good (95%) that you’ll have between 40 and 50 heads. With 1000 flips, the odds of being near 50/50 go up: 95% of the time you get between 468 and 531 heads. With 10,000 flips, they’re even higher: 95% of the time you’ll get between 4900 and 5100. In total, the number of heads approaches exactly 1/2 of the number of flips. But the chance of having longer and longer streaks also increases, and that’s what makes this so difficult to intuit. Because we experience these events through time, we asses the situation only as the information becomes available: the stock’s price history, or rather, return history, can exhibit several streaks, making it difficult for our pattern-recognizing brains to see the larger (mean reverting) picture. We also have a (very false!) expectation that the very next flip/price will bring the system back in line to where the average says it should be. In practice, we see reversion to the mean in a portfolio even though the individual stocks show no such behavior.

The analogy with coin flipping only goes so far – real securities aren’t independent of one another (there is correlation, sometimes quite high) and they do exhibit momentum effects, meaning that a given stock’s returns do in fact depend on the previous return. If coins behaved that way, a given flip would depend on what happened on the previous flip.

The fact is that mean-reversion is something that happens in aggregate – to the whole (portfolio, market, etc.), even when it doesn’t happen to the individual, whether that’s an individual coin flip or stock price. For me, there’s a little bit of magic in that. What do you think?