NEW YORK (TheStreet) -- So far in 2014, the Dow Jones Industrial Average
First we look at the theoretical distribution provided by a probability tree, in which each successive day has an equal 50% chance of either branching up or branching down. Then, for a specific tree path over the first seven days, a red value was given to indicate a down day, while a green value was given for an up day. For more on combinatorics, see this note.
We see that the first trading day for 2014 was down. And this had a 50% theoretical chance of happening. Then the second trading day was up, and so we end that day with a score of one down day out of two days. This score too has a 50% chance as shown (e.g., there is another 25% chance we'd instead have two up days in a row, and a 25% chance to instead have two down days in a row). Through this morning, Jan. 13, we've collected five down days out of seven trading days. We see on the tree above, that this outcome had a 16% chance of occurring. We also see that the probability of having five or more down days, out of seven trading days, is slightly larger at 23% (or ~16%+5%+1%).
A different interpretation of this 23% chance is that over the 14 years prior to this one, three of them (23%*14) should have also had at least 5 down days out of the first 7 trading days. And empirically this is true: 2001, 2005, 2009. That's some company.
But these probabilities were again derived assuming a fair 50% chance for either an up day or a down day. That indicates a close to trendless market, which lacks general direction, even though we technically understand that, over the long run, markets offer a slight upward trend. What would have been the probability of seeing five drops out of seven trading days, if we instead had assumed different up-day probabilities, ranging from 25%, to 75%? We see in the chart below, that as our assumption for the daily chance for an up day increases, the probability of having five drops (out of seven trading days) falls well below our 16% baseline.
Put differently, having five drops out of seven is less probable (from 16% to 6%) if we switch from assuming a fair 50% up-day probability to a slight up streak characterized by 60% up-day probabilities. Conversely, as the daily chance for an up day decreases, the probability of having five drops (out of seven trading days) rises well above our 16% baseline.
Let's then use our assumed probability for an up day to better explore our likelihood of outcome. We know that the chance that we are in a general phase that is far from "fair" (i.e., normal 50% probability of an up day) decreases the farther away the up-day probability is from 50%. Being in a strong up streak (e.g., 75% up-day probability) or in a strong down streak (e.g., 25% probability of an up day) for extended periods is simply not as likely versus being in a phase closer to a 50% up-day probability.
Technically the markets have a slight upward trend, so their up-day probability is only slightly higher than 50%.
So we rearrange our likelihood distributions accordingly, and better understand our recent outcome taking this behavior into account (e.g., using the variance of a compounded Bernoulli distribution). We initially stated that having two up days out of the first seven trading days is a 28% probability, but in the illustration below we can see how our new Bayesian likelihood levels look. It shows that we are instead closer to being in a slight down-streak period, characterized by up-day probabilities closer to 40% (less than a fair 50% chance of an up-day).
In other words we are more likely to be tilted in a slight down-streak (about 39% likelihood of being in a 40% up-day probability phase) vs. being tilted in a slight up-streak (about 9% likelihood of being in a 60% up-day probability phase).
Whether we consider the 6% fair chance to see our YTD results from about a 60% up-day probability (as in the prior chart), or a 9% chance to see it from with Bayesian conditions (as in the chart below), both probabilities are too low to fit with any interpretation that 2014's YTD performance is just a continuation of last year's upward fast ride.
Recall that we said that there was a 16% chance of seeing the Dow Jones Industrials drop on five out of seven trading days, if we first assumed that each up-day probability was 50%. But per the chart above, if we see these five down days (out of seven trading days), we can see that we are more likely in a period that is not characterized with a daily 50/50 chance of going up or down.
So instead of a 30% likelihood for a down day in a case where we have a 50% up-day probability, we instead suggest that 2014's YTD performance shows a 39% likelihood of being in a down streak, with only about a 40% probability of being up on any given day.
This article represents the opinion of a contributor and not necessarily that of TheStreet or its editorial staff.