N-tuple S&P Patterns Across Decades, 1950s to 2011

A.G. Malliaris and Mary MalliarisEuro Working FinancialGroup

May 3-5, 2012, Rome, Italy

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

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

Two-Day Patterns Across Decades

Three-Day Patterns

Four-Day patterns

Five-Day Patterns

Number of Ups in Two Days

Number of Ups in Three Days

Number of Ups in Four Days

Number of Ups in Five Days

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

The Forecast Decision

Training set

4-day

pattern Count

Up

through

Today

Forecast

DDUD 760 DDU U

DDUU 1050 DDU U

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.

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.

Number and Percent of Correct Forecasts on Validation Set

7 Day 6 Day 5 Day 4 Day 3 Day 2 Day 1 Day Rand

212 218 212 205 205 195 219 178

54.8% 56.3% 54.8% 53.0% 53.0% 50.4% 56.6% 46.0%

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

Decision Tree Results

Result Count PercentCorrect 10,165 67.38%Wrong 4,922 32.62%Total 15,087  

Result Count PercentCorrect 222 57.36%Wrong 165 42.64%Total 387  

Training Set

Validation Set

Up and Down Forecasts on the Validation Set

  Forecasted Down

Forecasted Up

Actual Down tomorrow

72 96

  51.064% 39.024%Actual Up tomorrow

69 150

  48.936% 60.976%

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 todaythe 7-day pattern [for example UDUUDDU]number of Up movements in the last three days.

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.