One of the first things I did when I fired up my Frankenstein’s monster was plot a recurrence map between equity returns and their lagged value. This is something every dynamical systems monkey will do. I did. In physics, we call it, “looking at the phase space.” If you can find an embedding dimension (in this case, a lag which creates some kind of regularity), you can tell a lot about the dynamical system under consideration. I figured plotting returns against their lags would be too simple to be interesting, but I was dead wrong. I saw this:

A high quality compass rose can be seen on Berkshire Hathaway preferred stock





I convinced myself that nothing this simple could be important (and that it went away with decimalization), and moved on to more productive activities, like trying to get indexing working on my time series class, or figuring out how to make some dude’s kd-tree library do what I wanted it to. I realized just today, this was a mistake, as other people have also seen the pattern, and think it’s cool enough to publish papers on. None other than Timothy Falcon Crack, bane of wannabe quants (and their employers) everywhere, was a coauthor of the first paper to overtly notice this phenomenon.

A slightly later epoch pre-decimalization

You can sort of see why this pattern would fade out with decimalization. If you’re trading in “pieces of 8” (aka 1/8ths of a dollar), returns which don’t neatly divide into 1/8ths will not be possible. In other words, there are only 7 prices between $20 and $21, as opposed to 100 like there are now. Therefore you’d expect to see some gaps in the lagged returns, which are just price ratios. Roughly speaking, if the average variance is small compared to the size of the tick, you’ll be able to see the pattern. At least that’s what most people seem to think. Weird that Berkshire Hathaway should be effected by this, but as it turns out, it had an effective tick size which was fairly large compared to daily motion, because the people trading it were lazy apes who wouldn’t quote a price at market defined tick size (which, even at 1/8ths was very small compared to Berkshire Hathaway’s share price of several tens of thousands of dollars).

Here you can still see some evidence of Compass Rose in the early decimalization era

One of the interesting implications of all this: if ticks are important enough to show up in a simple plot like this, what happens when you apply models which assume real numbers (aka virtually all models) to data which are actually integers? This is something I’ve wondered about since I got into this business. Anyone who notices his model returning something which has many decimal points at the end …. when the thing you’re measuring should be measured in integers should notice this. I don’t think this sort of issue has ever been resolved to anyone’s satisfaction; people just assume the generating process uses real numbers underneath, and average up to the nearest integer; sort of like trusting the floating point processor in your computer to do the right thing. The compass rose points out dramatically that you can’t really do that. It also demonstrates that, in a very real way, the models are wrong: they can’t reproduce this pattern. For example, what do you do when you’re testing for a random walk on something like this? Can it possibly be a random walk if the returns are probabilistically “loitering” at these critical angles? Does this bias models we use? Smart people think it does. Traders don’t seem to worry about it.

Finally, the compass rose is completely gone in the more recent epoch of decimalization for Berkshire Hathaway series A

Some other guys have attempted to tease some dynamics out of the pattern. Not sure I buy the arguments, since I don’t understand their denoising techniques. Others (Koppl and Nardone) have speculated that “big players” like central banks cause this sort of effect by creating confusion, though I can’t for the life of me see why central bank interventions would cause these patterns in equities. Their argument seems sound statistically. It was done on the Rouble market during periods of credit money versus gold backed. Unfortunately, they never bother relating the pattern in the different regimes to central bank interventions, other than to notice they coincidentally seem to happen at the same times. That doesn’t make any sense to me. It’s a regression on two numbers.

My own guess, developed over a half day of thinking about this and fiddling with plots in R, is that these patterns arise from dealer liquidity issues and market dislocations. How?

Human beings like round numbers. Machines don’t care. Lots of the market in ye olden pre-decimalization days was organized by actual human beings, like my pal Moe. Thus, even if there was no reason to pin a share at a round number, people often would anyway, because $22.00 is more satisfying than $22.13. Since liquidity peddling is now done by machines, most of which assume random walk, I’d expect compass rose patterns to go away in cases where it persisted for a long time, like with $100k Berkshire Hathaway preferred shares, which are all that is pictured above. Voila, I am right. At least in my one stock guess, though the effect can be seen elsewhere also. The plethora of machine-run strategies has made the market much more tightly coupled than it used to be. What does this mean? For example: at the end of the day, something like an ETF has to be marked to its individual components. One of the things which causes a burst in end of the day trading is the battle between the ETF traders trying to track an index, and arbs trying to make a dollar off of them. Similarly with the volatility of the index. With all this going on, there isn’t much “inertia” pinning the closing price to a nice, human round value. It was observed early on that indexes don’t follow the compass rose pattern, and it’s very easy to understand why if you think of it from the behavioral point of view; add together a lot of numbers, even if they’re mostly round numbers, and chances are high you will not get a round number as a result (especially if you weight the numbers, like in most indexes). You could look at the dissolution of this pattern over time as increasing the entropy of stock prices. High frequency traders make the market “hotter.” As such, the lovely crystaline compass rose pattern “melts” at higher temperatures, just like an ice cube in a glass of rum. With the Berkshire Hathaway preferred shares patterns above, you can see the pattern fading out as the machines take over: while some compass rose remains post-decimalization, it’s completely gone after 2006. You might see it at shorter time scales, however.

Relating it back to the Rouble analysis of Koppl and Nardone, I’d say they saw the compass rose in times of credit money simply because the market moved a lot slower than it did when it was based on gold. When it was credit money, there were effectively fewer people in the trade, and so, monkey preferences prevailed. When it was gold, there were lots of people in the trade, and the “end of day” for trading the Rouble was less meaningful, since gold was traded around the world.

One of the things that bothered me about the original paper is the insistence that one couldn’t possibly make money off of this pattern. I say, people probably were making money off the pattern: mostly market makers. What is more, I posit that, where the pattern exists (on whatever time scale), one could make money probabilistically. What you’re doing here is bidding on ebay. Everyone on ebay knows that it’s a win to not bid on round numbers, because the other apes will bid there. If you bid off the round number, you are more likely to win the auction. Similarly, if you’re a market maker, you might win the trade by bidding off the round number, and giving the customer a slightly better price. Duh. My four hours worth of hypothesis would predict thinly traded stocks which aren’t obviously important components in any index would continue to show this end of day pattern, since they won’t be as subject to electronic market making. And, in fact, that’s what I saw in the first one I saw, WVVI, which appears to be a small winery of some kind. Even in the most recent era, it has a decent compass rose evident. Second one I looked at, ATRO (a small aerospace company) similarly showed the compass rose during the 2001-2006 regime. I’m pretty sure there are simple ways to data mine for this pattern in the universe of stocks using KNN, though I don’t feel like writing the code to do it for a dumb blog post; someone’s grad student can look into it.





All of this is pure speculation after too much coffee, but it’s a very simple and evocative feature of markets which is deeper than I first thought. Maybe with some research one could actually use such a thing to look for trading opportunities (probably it’s just a bad proxy for “low volume”). Or maybe the excess coffee is making me crazy, and these patterns are actually just meaningless. None the less, in this silly little exercise, we can see effects of the integer nature of money, behavioral economics, visible market microstructure on a daily time scale, and very deep issues into the dynamics of financial instruments.