Random thoughts (or, rather, thoughts about randomness)

Let’s play a game.  Imagine we have before us the proverbial “fair coin”, and we decide to flip that coin 1,000,000 times.  When it comes up heads, I will pay you $1, and, of course, when it comes up tails you will pay me $1.  We will keep a running total of your “P&L” on every turn of this game.  Simple, right?  Please think about this question before you read further:

  • How much do you expect to make or lose at the end of this game?  (Come up with an actual dollar amount.)

This is obviously a game where neither of us has an edge, so the 20 days or so it takes us to play this game are totally wasted!  If you play this game enough times, you should break even on average (formally, this is a 0 expectancy game), but what may surprise you is that the odds of you having exactly $0 after a million trials is very, very, very small.  Much more likely you will be up or down a few hundred than exactly zero.  That question was pretty easy–here is a much harder one.

  • If we keep a running total of your net P&L on each turn, it seems obvious that it will cross zero sometimes.  (Imagine you win $1 on the first turn, then win $1 on the next, then lose on the next 2 and you are back at $0 total.  In this case you were positive for 3 turns.)  How many turns, in this random, breakeven game, would you expect to be positive or negative in a row?  In other words, if we were to graph your running total P&L for a million turns, how many turns in a row would you expect it to be above or below the zero line?  100?  1,000?  10,000?  More?
  • And a related question:  If we play this game for a million turns, do you think your P&L will cross zero many times, or is it more likely to spend a lot of time either positive or negative?

I built a quick computer simulation to give us some idea what might happen in this game.  I ran it 5 times for 1,000,000 coin flips each game.  I kept track of the number of heads that came up and the maximum number of flips your P&L stayed on one side of the zero line.  Last, I graphed your P&L for each of the 5 games.  I think the answers may surprise you.

For the five games, you flipped heads 50.03%, 50.01%, 49.99%, 50.05%, and 49.98%.  Nothing shocking there–pretty much just confirms that the coin was fair.  (As I said before, the chances of flipping exactly 50% heads is exceedingly small.)  Here is what the graph of your P&L for the first game of 1,000,000 flips looks like:

P&L record for 1 million coin flips

Wait a minute?  Really?  Most people expect that the P&L would stick pretty closely to the zero line, but, in reality, excursions from the line are VERY common.  Furthermore, once you are far away from zero, it is more likely you will spend more time away from it than go back.  In fact, in this test run (which really was the first one I did), there were 814,763 (out of 1 million) consecutive flips on the positive sign of the zero line!  Lest you think this run was an aberration, for the other tests, the numbers were 71,903, 646,124, 453,474 and 87,492 consecutive closes above or below zero.  What is going on here and why does this seem so strange to us?  It turns out there are some important lessons for traders here.

The bottom line is that humans tend to have poor intuition about random events.  We underestimate how “streaky” random data is, and so we have a difficult time looking at data and making a judgment about whether it comes from a random process or not.  In the graph above, your P&L was above zero for more than 800,000 flips of the coin, and, to a casual visual inspection, the chart appears to be trending.  If this was an actual system you were trading, do you think you would have concluded you had some actual edge in the market?

There is another lesson here.  Instead of being a graph of your P&L for my silly coin game, imagine that the chart above is a chart of a stock’s daily prices.  (In fact, what I have generated is one kind of “random walk” price path, so this is a legitimate way to think about it.)  Look at that chart (and the ones I will attach to the end of this post).  Do you see trends?  Do you see support and resistance?  Of course… look how well the “zero line” provided support twice, and the the stock traded away and never looked back.  If this is your first time looking at random price paths like this, are you surprised to see these patterns?  Remember, these charts really are truly random.  Any patterns we see are just due to random chance.  There is no actual support and resistance on these charts.  Every trend or trading range you see is purely the result of randomness.  There is nothing but randomness, but your eye will manufacture patterns in the same way that, presented with random patterns in clouds, you will see faces.

So what is the point here?  A reminder to be careful in your thinking. Don’t assume that , when something happens in the market, it “can’t be random.”  Random market action can produce patterns that might surprise you.  Don’t assume that, when you have a string of good trades, you necessarily have really found a sustainable edge in the market.  Many traders find it very uncomfortable and challenging to think about these topics, but I believe they are very important because they drive right to our core beliefs about what makes the market work.

(Take a look at the charts from the other four games.  Remember, each of these are 1 million flips of a fair coin, but evaluate them as if they were stock charts.  Do you see trends?  Trading ranges?  Support levels?  Hmmm….)

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