Irrational Numbers: Price Clustering & Stop Losses

Universal Number Theory

One of the odder things about the universe is that the small set of numbers that define its structure, the so-called universal constants, don’t seem to have any structure of their own. You’d have thought that whatever immortal deity breathed life into the whole shebang would have at least have bothered to make sure that reality was defined in simple integer values your average gameshow contestant could remember. Yet someone’s just calculated Pi to more decimal places than you can read in a lifetime. The universe is strangely irrational, it would seem.

More likely, however, is that the irrationality lies in our heads. If you look at the way we treat numbers for investment purposes it’s probably a good job the infinite cosmos is specified in irrational numbers, because if it were otherwise we’d probably have sold it to the lowest bidder eons ago. Humans, it seems, treat numbers as an approximation to reality, unlike reality; which treats humans as an approximation to nothing.

That Friday 13th Feeling

Under standard economic theories one price should be much the same as another but all experienced practitioners know that this isn’t so – some numbers are much more likely to occur than others. Anyone with even a basic appreciation of behavioural psychology would expect no more or no less – people are as arbitrarily inconsistent about numbers as they are about everything else. In western culture, for instance, thirteen has acquired negative connotations to the point where many tower blocks omit the number from their floor numbering plans, presumably on the grounds that the universe can’t count. Beware, for fourteen is the new thirteen. Ha!

Despite the obvious irrationality of ascribing luck to a number many people are petrified of Fridays falling on the thirteenth of the month. Such is the human propensity to translate mental muddle into actual behavioural nonsense that it turns out that more accidents do occur on these days. So either there’s a malevolent demon tripping us up or our incipient fears are causing us to fall over our own feet. Mental confusion in our heads often turns into real problems in the real-world.

Round Number Attractions

It’s no surprise to find this numerological naughtiness feeding across into investment, a discipline many stockmarket dabblers may be surprised to discover should involve a basic understanding of numbers and the use of simple mathematical operators (don’t tell them). In particular we should note that most humans, but not all, are addicted to round numbers and, because of this, securities prices tend to huddle together like waifs in a storm.

There are various effects associated with round numbers – especially those that end in zeros or, to a lesser extent, those ending in 5 or 25. So $1.00 is preferable to $1.50 and that to $1.25. They’re all preferable to $1.13, a figure that causes a goodly proportion of the population to start getting all touchy-feely with the nearest piece of tree-related product, lopping feet off innocent lagomorphs and chiselling shoes off any nearby loitering pack animals.

The Clustering Conundrum

This attraction to round numbers was first noted in research on price clustering – the tendency of prices to cluster around specific numbers rather than spreading themselves out randomly over the whole spectrum of possible values. As far back as 1962 Osborne showed that NYSE closing prices clustered around whole numbers and common fractions – halves and quarters, prior to decimalisation. Since then price clustering has been demonstrated time and again in all sorts of securities – stock indexes, stock prices, commodities, bonds, foreign exchange and futures.

Quite why this happens is a source of much debate, although the main theory is that it’s something to do with our cognitive limitations. Perhaps it’s a retrieval problem – the easy availability of round numbers to our mental processes makes them attractive. Perhaps it’s an anchoring problem, where investors unconsciously anchor on easily available round numbers.

Whatever it is, it has an effect on investment returns – Herrmann and Thomas (2005) have shown that financial analysts tend to round their forecasts and that market reaction to earnings surprises is based on these rounded numbers. Of course, the more cynical reader might view the rounding off of analyst forecasts as more evidence that these are one step removed from guesses, although a more generous view might be that all forecasts are rough estimates at best.

Beat The Markets

Johnson, Johnson and Shanthikumar have investigated the round number phenomena and have concluded that round numbers are implicated in stock movements – investors appear to trade differently when closing prices are just below a round number than just above. To whit, you get more selling in the former case and more buying in the latter. This doesn’t appear to be a naive trader or a technical analysis problem either; the result appears to be robust for institutional investors as well as private ones. Rather startlingly the researchers state:

“Our estimates show that returns following prices ending in 01 through 09 cents are, on average, 12.9 basis points higher than those ending in 91 through 99 cents. This corresponds to an annualized rate of over 38% per year....” (To be continued)

Now that’s what I call an effect.

Stop the Stop Losses

More evidence for this was uncovered by Joep Sonnemans, who looked at Dutch investor behaviour before and after the introduction of the euro in 1999. What he uncovered was that price clustering around round guilder prices disappeared overnight and turned into price clustering around round euro prices. As he states:

“Apparently, not only for a consumer 19.90 looks much less than 20.00, it also looks that way for an investor.”

Naturally this effect can be used against investors. The idea of using a stop-loss – a pre-set lower price limit at which securities are automatically sold – is a very common one, encouraged by many market practitioners. However, small traders are often heard to complain about being “stopped out” of a stock by an unexpected and short-lived downward spike in the price, bitterly breathing of conspiracy theories. Looking at the round number effect it’s pretty clear that you don’t need to invoke insider dealing to create this effect, you can rely on the traders' own behavioural biases.

Chinese Whispers

Price clustering around round numbers seems, therefore, to be a robust effect causing measurable effects. However, a number of researchers have wondered whether this is not just a cognitive limitation but a cultural effect. Brown and Mitchell used the multi-tier nature of the Chinese stockmarkets to investigate this. They hypothesised that if clustering is a cultural effect then the number eight, which is synonymous with good luck in China, would appear more often in prices than the number four, which is similar to the Cantonese word for death.

The structure of the Chinese markets was originally divided between A shares for Chinese local investors only and B shares for overseas investors only, although this distinction is slowly being eroded. By examining prices of the different classes of stock the researchers concluded that Chinese investors were being influenced by their cultural preferences for the number eight: opening, closing, high and low prices were far, far more likely to end in an ‘8’ than a ‘4’.

Controlling for the same features in western markets produced no such equivalent preference. In conclusion the researchers also observed that these cultural effects started to break down as external investors started to dominate, although the wider evidence might suggest that this was just replacing a local cultural preference with a wider behavioural bias.

Get Your Lunch Elsewhere

Clustering around round numbers is clearly not a function of any underlying valuation process, but is a cognitive simplification that we use to avoid complicating investment issues. In aggregate individual preferences for nice round numbers adds up to short-term irrationality as prices bounce above and below resistance points. All of which, rather startlingly, suggests that technical analysts may be on to something.

To rescue intelligent investing from the dustbin of humanity’s insatiable appetite for immediate gratification, however, we can conclude with the second part of the Johnson, Johnson and Shanthikumar quote from above:

“... It is unlikely that these return differences alone could be used to form a profitable trading strategy given the almost daily rebalancing required, but they may have a significantly positive impact on returns if exploited for optimizing the timing of order execution.”

Always remember the universe’s rule #3.14159265: no free lunches.


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