### Consecutive Up Days on the Stock Market and the Gambler’s Fallacy

Updated: Mar 10th, 2017 | Vance Harwood | @6_Figure_Invest

Years ago I was watching the action around an Atlantic City roulette table when I noticed a man taking careful notes.   When I asked what he was doing he said he was looking for long consecutive strings of red or black.  If the run got long enough he was going to bet that the ball on the next spin would land on the opposite color because it was “due.”

When I asked him how the ball knew what colors it had previously landed on he frowned and shifted a few steps away.

The Gambler’s Fallacy is an intuitive belief that long streaks, even with fair coins or dice, influence the odds of the next result.

On a day to day basis, the stock market is a lot like the game of roulette, with the probability of an up or down day being very close to 50%.  However unlike the roulette ball, the people (and computers) trading on the stock market can remember what has happened previously—so after a long stretch of up days, you would expect a higher probability of a down day.  Or so I thought.

For a long time I’ve believed that the market tends to move in 3-day cycles, but inspired by reading Moneyball (borrow free on Amazon prime), I decided to see if the data supported my intuition.   I took the S&P 500 index data from 1993 to 2013 and analyzed market moves after 3, 4, 5, and 6 up days in a row.   Rather than just plotting binary up or down results I plotted the frequency of the percentage results using 0.1% bins.

The red line shows the outline of what a perfect normal distribution would look like.  The actual data is more clustered around zero, under-represented on the sides of the distribution, with some rather extreme events in the tails of the distribution.

Instead of supporting my intuition the data shows that the stock market is very much like roulette, with no directional bias after three consecutive up days.  The average return (mean) is +0.016%— indicating very close to even odds.

I repeated the analysis for bull runs of 4, 5, and 6 consecutive days.

These sequences also show average returns close to zero.  With even odds the number of consecutive days with gains should decrease by 50% for each additional day—for example 6 day bull runs should happen half as often as 5 day runs.  And that is how the market behaves.   The chart below summarizes the statistical results.

The 5 Up Days statistics are distorted by a giant 8.9% drop on December 1st, 2008 so I include the 5 up data with and without that drop.

 3 Up Days 4 Up Days 5 Up Days 5 Up Days (minus 1-Dec-08) 6 Up Days Occurrences 390 188 93 92 46 Average Next Day Returns (Mean) -0.0156% -0.109% 0.0444% 0.0522% -0.0360% Next Day Standard Deviation 0.99% 0.88% 1.19% 0.75% 0.58% Largest Loss -4.9% -3.3% -8.9% -1.8% -1.6% Largest Gain 4.7% 2.1% 2.2% 2.2% -2.1%

If you exclude the “7 sigma” event in 2008 there is a trend towards less actual volatility the next day with longer bull streaks.   It might be the case that the people involved with the market become increasingly aware of the bull run and start behaving cautiously.  This trend would be no help for directional plays, and it’s hard for me to imagine a volatility play that could take advantage of a one-day lull.

Clearly my notion that the market moves in 3-day cycles was bogus, and the data suggests that any sort of directional analysis based on market history is just another example of the Gambler’s Fallacy.

Friday, March 10th, 2017 | Vance Harwood
• Lore

Great article Vance. However, why you combine a binary event (3 up days) with a discreet measurement of the performance? Wouldn’t it be more useful for trading purposes to compare, say 5-10% market returns, (maybe happening in a limited number of days) with successive performance?

The distribution curves you created are strongly leptokurtic, guess I would be very happy to see them in a casino, perhaps this could be a tiny trading advantage?

• Hi Lore, I used the binary events because that was the pattern I had personally followed. It would be straightforward to modify the analysis to trigger off percentage gains as you suggest, but I strongly suspect the results would be the same–randomness rules.

The leptokurtic distributions are interesting, but the fairly common high sigma events could blow up a lot of schemes that tried to profit from the typical behaviors.

— Vance

• Pingback: By: Lore | TravelSquare()

• This is quite old, but I wanted to make a comment. Something isn’t right here. If there is no bias and it is just as likely to go up as it is to go down, and by the same amount, then the market shouldn’t move anywhere, but it does.

• Hi Kir, Good question. It turns out there is a bias, but it’s small (+0.16%) at a daily level. This bias is the long term average return of the S&P 500. For sequences of small numbers of days as I discussed in this post it’s insignificant.

• It does strongly support the random walk theory of the markets. Although fresh news will modify the distribution at any given point in time.

It’s also possible that the impact of past values results in a longer term shift in probabilities, rather than an immediately obvious shift in the next day outcome. I feel that there are methodologies for testing for such time series patterns, but I can’t recall them off the top of my head.

• drake

Vance, has a similar analysis been performed on individual stocks? It seems straightforward and logical that a market index has no memory but I imagine the gambler’s fallacy has a more significant effect on individual stocks.

• frank furter

Tom DeMark’s indicators enjoy a remarkable endorsement in the financial industry. Researchers at ETH Zurich recently investigated the predictive power of three DeMark indicators over 21 commodity futures markets and 10 years of data. For the period from Jan. 2004 to Jan. 2014, the tests suggest statistically significant predictive power on a wide range of commodity futures.