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.