1. N-tuple S&P Patterns Across Decades, 1950s to 2011 A.G. Malliaris and Mary Malliaris Euro Working FinancialGroup May 3-5, 2012, Rome, Italy

2. Purpose To investigate the Up and Down movements of the S&P 500 from 1950 through 2011 Daily closing prices from 1/3/1950 to 7/19/2011, a total of 15,488 observations, were transformed into Up or Down by comparing today’s value to yesterday’s

3. Up [U] and Down [D] movements per decade, 1950 through 2011.

4. Two-Day Patterns Across Decades

5. Three-Day Patterns

6. Four-Day patterns

7. Five-Day Patterns

8. Number of Ups in Two Days

9. Number of Ups in Three Days

10. Number of Ups in Four Days

11. Number of Ups in Five Days

12. Forecasting Training set: data from January 1950 through December 2009 [15,087 rows]. With this data, we calculated the number of times each pattern occurred Validation set: January 2010 through mid- September 2011 [387 rows] Training set patterns were used to form predictions for the Validation set

13. The Forecast Decision Training set 4-day patternCount Up through Today Forecast DDUD760DDUU DDUU1050DDUU Suppose that DDU has occurred. Past history tells us that, of the four day possibilities, DDUU is more likely to occur than DDUD. So, for tomorrow, we will forecast Up when the three days before tomorrow have the pattern DDU.

14. Random Forecast We compare the performance of this conditional forecast of n-tuples with the random walk forecast. A random number was generated If the value was less than.5 then Down was predicted, otherwise the forecast was Up.

15. Number and Percent of Correct Forecasts on Validation Set 7 Day6 Day5 Day4 Day3 Day2 Day1 DayRand 212218212205 195219178 54.8%56.3%54.8%53.0% 50.4%56.6%46.0%

16. Decision Tree Model A C5.0 Decision Tree was built using IBM’s SPSS Modeler 14 package. We gave the decision tree the following inputs to use in building the tree: the up-down patterns from one to seven days the number of up days in 1 to five days and the closing value today

17. Decision Tree Results ResultCountPercent Correct10,16567.38% Wrong4,92232.62% Total15,087 ResultCountPercent Correct22257.36% Wrong16542.64% Total387 Training Set Validation Set

18. Up and Down Forecasts on the Validation Set Forecasted Down Forecasted Up Actual Down tomorrow 7296 51.064%39.024% Actual Up tomorrow 69150 48.936%60.976%

19. Variable Importance This Modeler technique also ranks the input variables in terms of importance, with more important variables occurring higher up on the tree. The variables ranked highest in importance to the forecast were the direction today the closing value today the 7-day pattern [for example UDUUDDU] number of Up movements in the last three days.

20. Conclusions The number of up movements is greater than down movements across the decades. Among the seven n-tuple lengths that we studied, the highest ratio of successful forecasts was the one conditioned on only today’s direction to predict tomorrow. Markets do seem to have an upward trend Successful trading can be achieved on the very short-term basis of one day.