It's happening again now, for the second year straight. There have been 18 Tuesdays, when the markets have been open, so far in 2014. And 15 of those ended in up-days. While we can expect to see some pattern away from the mean, 15 is well beyond reasonable due to random market fluctuations. What's worse, this is the same abberative pattern we saw at the start of 2013: clocking in 15 up-days of the first 18 Tuesdays. This note makes clear how rare such an event would be, through normal chance. We leave with a compelling case that something instead is also behind this strange Tuesday phenomenon. It is not merely (or even mostly) to be ignored as some fascinating, quirk of nature.
If we model the markets as something based off of a normal, random walk, then we should expect that just over half of the trading days would be up-days. Such as flipping a coin: heads the markets are up, and tails the markets are down. There should not be any more serious rhyme or reason beyond that. But let's look at the start of the year recently, a period defined here as through May 7. This period is not two or three trading Tuesdays. It is 18 trading Tuesdays, in each recent year, from 2011 through 2014.
Now 2011 and 2012 are fine, with both having roughly just over half their Tuesdays as up-days. But things are different now, first in 2013, and now in 2014 as well. We see in the chart below the purple probability associated with the normal rate of up-days that we saw in 2011-2012, and we contrast it with the super abberative rate we are seeing now in 2013-2014 (one might be able to click on chart below to magnify it).
If we model the markets as something based off of a normal, random walk, then we should expect that just over half of the trading days would be up-days. Such as flipping a coin: heads the markets are up, and tails the markets are down. There should not be any more serious rhyme or reason beyond that. But let's look at the start of the year recently, a period defined here as through May 7. This period is not two or three trading Tuesdays. It is 18 trading Tuesdays, in each recent year, from 2011 through 2014.
Now 2011 and 2012 are fine, with both having roughly just over half their Tuesdays as up-days. But things are different now, first in 2013, and now in 2014 as well. We see in the chart below the purple probability associated with the normal rate of up-days that we saw in 2011-2012, and we contrast it with the super abberative rate we are seeing now in 2013-2014 (one might be able to click on chart below to magnify it).
The initial probability distribution starts with a fair baseline of a 50% up-day probability for any given day. In this baseline assumption, there is a 0.3% probability of seeing exactly 15 up-days of 18. In parenthesis beside that statistic, we show there is only a total of 0.4% probability of instead seeing a minimum of 15 up-days of 18. So this could include any of these number of up-days for the start of any given year: 15, 16, 17, or 18.
We can now repeat this process by enlarging the up-day probability assumption to: 55%, or we can aggressively go as high as 60%, or 65%.
On this chart, pay attention to the orange colored row. It shows that even if we assume a general up-tilt in the markets (relative to the baseline), we still typically see low probabilities associated with seeing 15 up-days of 18 for any given year.
Now another probability consideration for this combinatorial math, above, is to also approximate the solution here through confidence intervals and the Student-t distribution. We know that there is a standard deviation about our up-day estimate, which inversely related to the root of the dof. Where dof equals degrees of freedom for our sample. From this estimate we get about a 1% chance of seeing a minimum of 15 up-days of 18, for the case of the 50% up-day probability baseline. We get in the middle of a 5%-10% chance range, for the case of an enlarged 65% up-day probability scenario. While these chances are a tad higher, given we have a modest sample of 18, the values are still completely aligned to the parenthesized orange values on the chart above. For allied research on the confidence interval of proportion estimates, see this article.
Before going further, let’s outline three samples of previous research articles where we discuss the general probability theory that we use here (and in each case we show in parentheses afterwards how this note is uniquely different). The first note covers the chance
of seeing any weekday having a x
up-day streak (in this note today we look at a specific workday’s chances and
not primarily for a streak). On an aside, this previous note was done almost exactly a year ago, when this same pattern was spotted. Then Barron's cited a Statistical Ideas article on how the S&P implied volatility gauge was also picking up this pattern. Returning to today's combinatorial topic, a second
note discusses the probability of not
seeing a down streak of z
days out of y months (in this note today we are not looking at a down
streak). A third note
discusses the probability associated with seeing z down-days out of y total days, including successive
patterns within them (in this note today we focus generally on minimal up-days instead
of the probability of down-days).
We know that the pattern that we've seen thusfar, in the up-day frequencies, imply that we are unlikely to seek an analogy similar to a fair and identical coin toss (see this Bayesian article). Let’s also explore the typical
returns, decomposed between up-days and down-days, to get a sense of severity as well from 2011-2014.
So we also see that the underlying characteristics of the Tuesday
returns, during 2013 and 2014, are planted to the up-side, as opposed
to the down-side. Combined with the tilt in up-day probability, we are seeing a greater case for focusing on this issue of Tuesdays, now in the early part of the year.
Let's re-portray the probability of seeing 15 up-days of 18, but instead in a common form of an up-day streak. This can help up better understand the chance of seeing the current Tuesday pattern. We show further the number of up-day streaks that are required to best match the probability associated for 2011-2012, and 2013-2014. In both cases these are the probabilities in parentheses in the top-chart above. For each up-streak we maintain the probability of an up-day, for any given day, is the same respective 50%, 55%, 60%, and 65%. The result is that the pattern this year is equivalent to about an up-day streak of 8, with a 50% up-day probability in any given day. Or the equivalent to about an up-day streak of about 6, with a 65% up-day probability in any given day. For the case of about a 60% up-day probability, we have a streak of roughly 7. Imagine tabulating slightly unfair coin flips from your right hand, where a coin flip result of heads (~60% probability) is represented by H, and of tails (~40% probability) is represented by T:
right hand flips: H-H-H-H-H-H-H
Finally we note that the results we saw in 2011-2012 were so typical, that they generally equate to not seeing an exciting streak whatsoever.
right hand flips: H-H-H-H-H-H-H
Finally we note that the results we saw in 2011-2012 were so typical, that they generally equate to not seeing an exciting streak whatsoever.
In addition to changing a given year's assumption for the true up-day probability
(not to be confused with the non-utility based, risk-neutral up-day probability), let’s look at
another market dynamic that can also at times help explain our recent 15 up-days of 18. In a normal random walk we can typically expect no autocorrelation in returns, particularly for higher-frequency data.
But let’s now instead introduce a specific serial relationship.
We enhance the odds for a current given Tuesday’s direction, based on the previous Tuesday's direction. And below we see what the basic probability distribution of up-days would be, using the 50% up-day probability baseline in the top-chart above. Later we can assume a mild blending of both this up-day probability, and the natural gyration in auto-correlation values.
We enhance the odds for a current given Tuesday’s direction, based on the previous Tuesday's direction. And below we see what the basic probability distribution of up-days would be, using the 50% up-day probability baseline in the top-chart above. Later we can assume a mild blending of both this up-day probability, and the natural gyration in auto-correlation values.
Let's explore how the number of up-days of 18 vary, based on different autocorrelation assumptions. We start with our baseline assumption (on the top-most chart) of 0.00
autocorrelation. We then chart the impact of increasing this autocorrelation to 0.25 and 0.50. For completeness we continue to ramp up to extreme levels of 0.75
and 1.00. Using our open-form solution, which is a fine way to model this case with autocorrelation, we see that for autocorrelation of 0.50 or less, seeing 15 up-days of 18 is still a rarity.
The combination of the four data charts above show us some revealing information about the probability of seeing 15 up-days of 18, as we've seen for Tuesdays so far in 2014. The combination of common assumptions concerning market tilt and momentum, mixed with autocorrelation, still show the market that there is about a 5% chance of seeing 15 up-days of 18 for any given day. Of course the probability of seeing this, two years straight, is more complicated for random walk and fair-market enthusiasts. First the probability of seeing this level of up-days, in the just the start of the past two years, could justify about a 1% chance of occurring. Not a high probability at all. Though not so low to set off alarm.
The overall probability of seeing this over long periods of time is that we can not assume the generous handicaps of autocorrelation and up-tilt bias in each of all general, long-term probability calculations. We must revert back to something similar to perhaps 2% for any given year. And this would mean the recent two-year pattern (i.e., 15 up-days of 18), which we've discussed in this note that we are seeing on Tuesdays, could apply for any given day. However even with highly generous up-day assumptions, this is still a semimillennium (2 in a 1000) event! This early-year phenomenon would be too infrequent to explained by common market vicissitudes; what we're seeing now in 2013-2014 would be of alarm. We can rearrange the streak probabilities above, and think of this not with a one-handed, slightly unfair coin flip but jamming a second similar coin in the left fist to also flip simultaneously. This time with about 6 paired flips (instead of 7 single flips) with the following result:
right hand flips: H-H-H-H-H-H
left hand flips: H-H-H-H-H-H
Imagine how you would enjoy being forced into a locked room, until you can produce this result above. Seeing the market's produce back-to-back, 15 up-days of 18 is exceptionally rare. This phenomenon is not an accident of chance to be brushed aside. And the rarity calculations already include, per above, the probability modeling for common market gyration relationships - in order to help explain the aberration. Based on recently history, this pattern in up-days is one that continues through mostly Memorial Day, before dissipating as it should for the balance of the year. We have to seriously entertain the discussion that something (yet unexposed) is unduly impacting the Tuesday market performance in the early part of the year, and equally important can impact market psychology if shown to be the result of unfair manipulation.
The overall probability of seeing this over long periods of time is that we can not assume the generous handicaps of autocorrelation and up-tilt bias in each of all general, long-term probability calculations. We must revert back to something similar to perhaps 2% for any given year. And this would mean the recent two-year pattern (i.e., 15 up-days of 18), which we've discussed in this note that we are seeing on Tuesdays, could apply for any given day. However even with highly generous up-day assumptions, this is still a semimillennium (2 in a 1000) event! This early-year phenomenon would be too infrequent to explained by common market vicissitudes; what we're seeing now in 2013-2014 would be of alarm. We can rearrange the streak probabilities above, and think of this not with a one-handed, slightly unfair coin flip but jamming a second similar coin in the left fist to also flip simultaneously. This time with about 6 paired flips (instead of 7 single flips) with the following result:
right hand flips: H-H-H-H-H-H
left hand flips: H-H-H-H-H-H
Imagine how you would enjoy being forced into a locked room, until you can produce this result above. Seeing the market's produce back-to-back, 15 up-days of 18 is exceptionally rare. This phenomenon is not an accident of chance to be brushed aside. And the rarity calculations already include, per above, the probability modeling for common market gyration relationships - in order to help explain the aberration. Based on recently history, this pattern in up-days is one that continues through mostly Memorial Day, before dissipating as it should for the balance of the year. We have to seriously entertain the discussion that something (yet unexposed) is unduly impacting the Tuesday market performance in the early part of the year, and equally important can impact market psychology if shown to be the result of unfair manipulation.
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