28,000. The President's favored report card (the Dow) just
breached that milestone in the recent day's trading. Surging 223
points, or 0.8%. At such an elevated plateau,
it clearly doesn’t take much for the Dow to have a noticeable change. The typical daily drift, for example, is +0.04%: or just over 9 points using today’s
level. And the typical change for the Dow is +1%
(up or down) from the trend: or about +300 points in any given day. But what about simply 0̸? Or also known as zero change (or “UNCH”)?
Could such a precise change occur? Yes, and in fact it occurs more often than you might expect. In just the past couple weeks, we have had both a near-UNCH (e.g., at just a -0.07 points daily move, or -0.0003%), as well as an perfect UNCH. The Wall Street Journal had appreciated such uncanny rarity, and the intrigue spilled over to a front-page article.
I was cited in this article as well, explaining an UNCH was a less than once a decennial event, according using the best probability methods we have. Probability problems must start by being grounded in theory, and then we can reconcile to aberrations in any imprecise empirical data. And our citation in the Wall Street Journal was no different.
But through that process, there emerges a number of exciting questions that mathematicians would want to explore. Staring at the mystical topic of nothingness! For example, why can such a faint reading even occur? What is the empirical distribution about zero, to see the sensitivity of results? And last, how deviant of a finding should UNCH be, relative to what we know about underlying theory of the Dow's properties? This article confronts these inter-related queries above.
Factors that cause more UNCHs days
Let’s first discuss some of the mathematical properties of the Dow statistics, and how some shifts in parameters would cause more UNCH daily Dows, all else equal. The first is that trading algorithms, which trade derivative instruments correlated with the Dow’s 30 stock components, can create patterns that are not random. Their programmers themselves are not random, and instead follow and overlap one another in ideas. Their codified thinking also overlaps in accepted formulaic values, with unconsciously clusters around popular intervals.
The result is that UNCH is actually the most popular daily “change” of all! Occurring 18 times since 1985 (past 34 years). To put this in some perspective, +0.5 points is one of the second most popular changes over the same history. But that outcome is from a net 1/8 dollar fraction unit, used for pricing pre-decimalization.
Now another factor is the concept of excess-kurtosis (or fat-tail), which currently plays only a slight role. Yet it is not commonly understood that even the most robust fat-tail models are also fat-centered as well! Though generally not by that much. It’s only around the typical changes (e.g. about + 1% daily change) where there is lower-than-normal probability (link, link).
Another factor that can creep up from time to time, and plays a supporting role, is the randomness of market news flow. How each day the component volumes are both random, and unequal.
While it may appear that overall volatility would seem to play a role (seeing that the VIX is currently at 12%), this is not true at all. The VIX has even lower during 2017, as I was cited discussing in the NYT print, a year the Dow had no UNCH. See the Appendix for supplementary charts.
In other words, like for much of market history. Even two UNCH days could be quite dissimilar in many other respects. And no particular random explanatory aspect can be the cause of the UNCH.
Factors that can cause fewer UNCHs
Let’s now discuss some mathematical properties of the Dow statistics that would cause fewer UNCH days, all else equal. The first is periods of unequal component prices. If one stock (e.g., Boeing) is 10x the price of another component stock (e.g., Pfizer), then there needs to be 10x % change between the two company stocks, in opposite directions. Just for those two stocks to cancel one another, and help guide a Dow sum to be UNCH.
The second factor is that there is discretionary voting of component company changes. These changes factor a more precise Beta, and mathematically prevent long-term comparability of probability modeling to access the likelihood of a specific outcome. Recall that when the margins of error increase in statistical data (similar to election polls), one must add a statistical penalty in order to obtain an unbiased solution.
And third is that over time we have a rising Dow, for many economic and corporate engineering reasons. As alluded to before, a larger base implies an even more narrow % chance that the deviations would come out to be so small. Particularly when accounting for the progressively shrinking divisor (which over the recent decades dropped below 1, and is right now 0.14744568353097, near the 0.14 level it has been for much of the past decade). This cratering divisor is a fourth factor, which only magnify the results of the Dow sum and makes it more difficult to compute a precisely 0̸.
What’s in the empirical data
We see with the current divisor, that the most miniscule change (sans 0̸) is just 6 or 7 cents: or +0.0003%. This would could come from a single Dow component stock having a change of $0.01. Or, the sum of 30 stocks having this same $0.01 net change. Now, let's look at the chart below to see how the historic count of these multiplied sums near 0̸.
It should be quite obvious from this empirical distribution, that something is naturally amiss. To start, there are far more UNCH days than we’d expect from a normal
distribution, or even a higher-kurtosis distribution.
To make matter worse, at nearly +$0.07 we’d expect nearly a handful of occurrences in the history. And right now we have none, and only 3 on the other direction at about -$0.07. So much for the more muted near 0̸, from the larger divisors of the past. And so much for the positive market drift and symmetry we would expect from financial theory! This chart simply destroys the idea that natural theory would holds up at level zero.
Also note that sometimes we must "kernelize" the clusters that are near one another. For example, -$0.08 display is simply a rounding function of the divisor over time. It is essentially meant to be included with -$0.07 instead.
We clearly have a case where there is not much of a high-kurtosis impact (again very few counts at +$0.07), but rather the UNCH is soaking up much of the just neighboring near-UNCH counts. In any reasonable excess kurtosis scenario, there should be a much more flat/even distribution across the entire range (shown here from about -$0.2, to about $0.2... or up/down by just 0.002%!) We can even remove the impact of smaller Dow levels in the past, as this pattern holds in each decade since.
Again, this doesn’t continue forever in either direction, as we've noted at many units about the drift (roughly 10 points), there are local modes in these historical counts, which are better aligned to a normal distribution.
Appendix
Finally, we leave you with few illustrations that we think are noteworthy. In the initial chart below, we focus on the event of both an UNCH and near-UNCH (i.e., defined as within +$0.20 but excluding UNCH itself) over the Dow’s history since 1985.
While I was cited numerously testifying a 0̸ Dow would occur as a decadal event going forward, we see it happening more often than that. For example, we see it occurred 4 times this millenium so far. Additionally we've had our share of near-0̸ as well. The most recent was just a couple weeks ago, so close to the 0̸ from November 12 that the two marked dots are essentially overlapped on the upper-right of the chart above!
Could such a precise change occur? Yes, and in fact it occurs more often than you might expect. In just the past couple weeks, we have had both a near-UNCH (e.g., at just a -0.07 points daily move, or -0.0003%), as well as an perfect UNCH. The Wall Street Journal had appreciated such uncanny rarity, and the intrigue spilled over to a front-page article.
I was cited in this article as well, explaining an UNCH was a less than once a decennial event, according using the best probability methods we have. Probability problems must start by being grounded in theory, and then we can reconcile to aberrations in any imprecise empirical data. And our citation in the Wall Street Journal was no different.
But through that process, there emerges a number of exciting questions that mathematicians would want to explore. Staring at the mystical topic of nothingness! For example, why can such a faint reading even occur? What is the empirical distribution about zero, to see the sensitivity of results? And last, how deviant of a finding should UNCH be, relative to what we know about underlying theory of the Dow's properties? This article confronts these inter-related queries above.
Factors that cause more UNCHs days
Let’s first discuss some of the mathematical properties of the Dow statistics, and how some shifts in parameters would cause more UNCH daily Dows, all else equal. The first is that trading algorithms, which trade derivative instruments correlated with the Dow’s 30 stock components, can create patterns that are not random. Their programmers themselves are not random, and instead follow and overlap one another in ideas. Their codified thinking also overlaps in accepted formulaic values, with unconsciously clusters around popular intervals.
The result is that UNCH is actually the most popular daily “change” of all! Occurring 18 times since 1985 (past 34 years). To put this in some perspective, +0.5 points is one of the second most popular changes over the same history. But that outcome is from a net 1/8 dollar fraction unit, used for pricing pre-decimalization.
Now another factor is the concept of excess-kurtosis (or fat-tail), which currently plays only a slight role. Yet it is not commonly understood that even the most robust fat-tail models are also fat-centered as well! Though generally not by that much. It’s only around the typical changes (e.g. about + 1% daily change) where there is lower-than-normal probability (link, link).
Another factor that can creep up from time to time, and plays a supporting role, is the randomness of market news flow. How each day the component volumes are both random, and unequal.
While it may appear that overall volatility would seem to play a role (seeing that the VIX is currently at 12%), this is not true at all. The VIX has even lower during 2017, as I was cited discussing in the NYT print, a year the Dow had no UNCH. See the Appendix for supplementary charts.
In other words, like for much of market history. Even two UNCH days could be quite dissimilar in many other respects. And no particular random explanatory aspect can be the cause of the UNCH.
Factors that can cause fewer UNCHs
Let’s now discuss some mathematical properties of the Dow statistics that would cause fewer UNCH days, all else equal. The first is periods of unequal component prices. If one stock (e.g., Boeing) is 10x the price of another component stock (e.g., Pfizer), then there needs to be 10x % change between the two company stocks, in opposite directions. Just for those two stocks to cancel one another, and help guide a Dow sum to be UNCH.
The second factor is that there is discretionary voting of component company changes. These changes factor a more precise Beta, and mathematically prevent long-term comparability of probability modeling to access the likelihood of a specific outcome. Recall that when the margins of error increase in statistical data (similar to election polls), one must add a statistical penalty in order to obtain an unbiased solution.
And third is that over time we have a rising Dow, for many economic and corporate engineering reasons. As alluded to before, a larger base implies an even more narrow % chance that the deviations would come out to be so small. Particularly when accounting for the progressively shrinking divisor (which over the recent decades dropped below 1, and is right now 0.14744568353097, near the 0.14 level it has been for much of the past decade). This cratering divisor is a fourth factor, which only magnify the results of the Dow sum and makes it more difficult to compute a precisely 0̸.
What’s in the empirical data
We see with the current divisor, that the most miniscule change (sans 0̸) is just 6 or 7 cents: or +0.0003%. This would could come from a single Dow component stock having a change of $0.01. Or, the sum of 30 stocks having this same $0.01 net change. Now, let's look at the chart below to see how the historic count of these multiplied sums near 0̸.
To make matter worse, at nearly +$0.07 we’d expect nearly a handful of occurrences in the history. And right now we have none, and only 3 on the other direction at about -$0.07. So much for the more muted near 0̸, from the larger divisors of the past. And so much for the positive market drift and symmetry we would expect from financial theory! This chart simply destroys the idea that natural theory would holds up at level zero.
Also note that sometimes we must "kernelize" the clusters that are near one another. For example, -$0.08 display is simply a rounding function of the divisor over time. It is essentially meant to be included with -$0.07 instead.
We clearly have a case where there is not much of a high-kurtosis impact (again very few counts at +$0.07), but rather the UNCH is soaking up much of the just neighboring near-UNCH counts. In any reasonable excess kurtosis scenario, there should be a much more flat/even distribution across the entire range (shown here from about -$0.2, to about $0.2... or up/down by just 0.002%!) We can even remove the impact of smaller Dow levels in the past, as this pattern holds in each decade since.
Again, this doesn’t continue forever in either direction, as we've noted at many units about the drift (roughly 10 points), there are local modes in these historical counts, which are better aligned to a normal distribution.
Appendix
Finally, we leave you with few illustrations that we think are noteworthy. In the initial chart below, we focus on the event of both an UNCH and near-UNCH (i.e., defined as within +$0.20 but excluding UNCH itself) over the Dow’s history since 1985.
While I was cited numerously testifying a 0̸ Dow would occur as a decadal event going forward, we see it happening more often than that. For example, we see it occurred 4 times this millenium so far. Additionally we've had our share of near-0̸ as well. The most recent was just a couple weeks ago, so close to the 0̸ from November 12 that the two marked dots are essentially overlapped on the upper-right of the chart above!
Now in this chart above, we show the lack of connection between the daily Dow changes (at or near 0̸), and the VIX. We've had days in history for example where the VIX was north of 25%, and yet the Dow was perfectly 0̸ on the day! And of course, never did we have a 0̸ daily Dow on the small portion of history where the VIX was at 10% or less. And last, thought it would be splendid to share with you the cover of the Wall Street Journal this week
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