Perfect 10

Stocks, of course, continue their hot streak. With gains today, the Dow Jones Industrial Average has now been up 10 days in a row. The titled "Dow Winning Streak Captivates Traders, Tests Statisticians" appeared on Yahoo Finance's Breakout segment.

By now you've surely heard that the nine-day winning streak on the Dow Jones Industrial Average (^DJI) is its longest since 1996. If it succeeds today in extending its run for a tenth day, it would be the 25th time it has achieved this feat in the past 70 years.

 

To gain the all-time longest streak title, the Dow Industrials will have to topple a 14-day streak set back on June 14, 1897, according to Rebecca Patterson of S&P Dow Jones Indices. The longest modern day streak was 13 days in January 1987.

Even though I am not a statistician, I would like to take a crack at the statistical problem of market streak. We know that on any given day, there is about a 2/3 probability of stocks being up and 1/3 probability of stocks being down.  We may ask given such probabilities, what are the odds of a 10 day up streak in a given year.

Since stocks go up 2/3 of the time on a given day, then the probability of stocks going up 10 days in a row should simply be (2/3)^(10), or 1.73% of the time. Let us also assume that there are 300 trading days in a year as a result, we have 291 separate 10 day periods. During the first 10 trading days, we know there is a 1.73% of chance stocks will all appreciate. However, if that fails to happen, it may then happen on the second 10 day period or from day 2 to day 11. The probability of that is the probability of streak failing on the first 10 days which equals to 1-1.73% or 98.27%, multiplied by the probability of 10 up days in a row or 1.73%. This yields 1.70%. Of course, if both of these scenarios don't work out, the streak could happen on the third 10 day period or from day 3 to day 12. The probability of that is the probability of first two scenarios both failing at 1-1.73%-1.70% or 96.57% multiplied by the probability of 10 up days in a row. So the odds of day 3 to 12 producing a streak after day 1-11 have failed is 1.67%. Thus, we can recursively compute the probability of all 291 10-day periods after all the proceeding periods have failed. By adding up all those possibilities, we can arrive at the probability of a 10 streak within a given year.

By my computation, the odds of a 10 day up streak for any random year is 99.38%.

Most readers are likely surprised by such high odds of streak. I recall reading a professor (sorry for the lack of source) who asked his students to toss a coin 10 times in a row and record their results. Those students who genuinely did the experiments were far more likely to record streaks of 3 or more in their data. Those who fudged their data most often did not contain such streaks. Rare events will occur given enough trials such as the same lottery number appearing consecutively. The fact that most people under-estimate such occurrences is the reason why this world in general and stock market in particular are full of myths and superstitions.

The Dow Jones Industrial Average only had 25 such 10 day streaks over the past 70 years, which works out to be a probability of only 35%. Furthermore, the longest streak is only 13 days. This to me indicates that stock returns are in fact serially correlated. What happened yesterday does have an impact on what happens today, which will impact what will happen tomorrow.