Stock Direction Correlation & Distance Correlation

Comparing Up/Down movements of successive days in the FTSE gives us this unexpected result. Given today's close relative to yesterdays as Up/Down what what the previous move?

Yesterday was Down: Today was:

Up 1673 25.0% 52.5%

Down 1513 22.6% 47.5%

Total 3168 (of total) (of down)

Yesterday was Up: Today was:

Down 1673 25.0% 47.6%

Up 1839 27.5% 52.4%

Total 3512 (of total) (of up)

There are only 18 days when the two days close the same i.e. 0.3% so I'm ignoring this event.

I don't need to do stats on this: there are 6698 events! and unbelievably the Up/Down and Down/Up counts are the same really suggesting that it makes no difference whether yesterday was an up or a down to signal a change in direction: it happens 25% of the time i.e. the Expected value based upon a random assumption.

However if yesterday was Up then there is a more than average chance that today will be up also! 27.5%.

Conversely if yesterday was down there is a less than average chance that tomorrow will be down!

In the stock walk then, the stock has a 52% chance of going up again and 48% chance of falling. Does this explain the trend of the stock market up or is distance a feature also, that is, are moves down bigger than moves up or vice versa.

The total log distance moved in the period is from 7.010402 on the 4th Feb 1984 to 8.642715 on 26th Nov 2010 which is +1.63231. There are 3522 up events (52.4%) in total and 3194 down events (47.6%) i.e. an excess of 328 up events to account for the up shift, if up and down are the same size. So we would expect each event to be on average a shift of 0.5%.

We can calculate that exactly from the data. The average move irrespective of direction is 0.7922% much bigger. This breaks down to:

Average move Up is: 0.7805% against an average move down of -0.81%.

So to summarise the probability of up moves is 10.3% more likely than down moves, but the size of down moves is 3.7% larger than up events. The average daily swing is also larger at 0.8% than the minimum expected at random of 0.5%.

A Model

N = number days -1. Pu/Pd is probability of up/dow

n. Du/Dd is average distance up/down.

Total distance Up = N.Pu.Du

Total distance Down = N.Pd.Dd = N.(1-Pu).Dd

Net distance D = Total distance Up - Total distance Down = N [Pu.(Du+Dd) - Dd)]

Our data set has N=6735 interday events. Pu = 0.524. Du = 0.0078. Dd= 0.0081

D = 1.56

Data for BP dataset.

Yesterday was Down. Today is:

Up 1640 25.5

Down 1486 23.1

Yesterday was Up. Today is:

Down 1585 24.7

Up 1717 26.7

There are 4.5% events when the stock closed the same. Probably a feature of the lower stock value and therefore less significant figures and then rounding errors. Ignored.

There is a similar pattern to the FTSE. Noticeably the probability of an Up is higher if yesterday was a Down than the other way around. I need to think about this in relation to the extraordinary result from the FTSE.

Average distance Up is: 0.012396

Average distance Down is: -0.0124

Total average distance: 0.011868 (including days when stock same)

Interestingly unlike the FTSE the average moves up and down are the same. This means the trend over time is because of the probability of up/down rather than the size!!! Very interesting from a trading point of view!! (Maybe individual stocks work like this while the FTSE is a conglomeration of different sectors each with different performances each day and so has changing behaviour.)

The Model is a bit simpler here:

N = 6428, Pu = 0.514, Du = 0.012

D = N [Du(2.Pu - 1)] = 2.16 (Actual value of : 2.28)

Next up is there a correlation between distances?

Not Obviously but stats tests show otherwise. A two tailed on the complete data of today's price change against tomorrow's was less conclusive with the Pearson not even being significant.

Correlations
		Today	Tomorrow
Today	Pearson Correlation	1	.008
	Sig. (2-tailed)		.534
	Sum of Squares and Cross-products	1.927	.015
	Covariance	.000	.000
	N	6733	6733

The Spearman and Kendal-tau measures show a significant correlation between todays and tomorrows stock moves.

Correlations
			Today	Tomorrow
Kendall's tau_b	Today	Correlation Coefficient	1.000	.022**
		Sig. (2-tailed)	.	.008
		N	6733	6733
	Tomorrow	Correlation Coefficient	.022**	1.000
		Sig. (2-tailed)	.008	.
		N	6733	6733
Spearman's rho	Today	Correlation Coefficient	1.000	.031*
		Sig. (2-tailed)	.	.010
		N	6733	6733
	Tomorrow	Correlation Coefficient	.031*	1.000
		Sig. (2-tailed)	.010	.
		N	6733	6733
**. Correlation is significant at the 0.01 level (2-tailed).
*. Correlation is significant at the 0.05 level (2-tailed). However taking absolute values and so only looking at the size of the change the correlations are not surprisingly extremely significant. Today_pos Tomorrow_pos Today_pos Pearson Correlation 1 .210** Sig. (1-tailed) .000 Sum of Squares and Cross-products .979 .205 Covariance .000 .000 N 6733 6733 Correlations Today_pos Tomorrow_pos Kendall's tau_b Today_pos Correlation Coefficient 1.000 .067** Sig. (1-tailed) . .000 N 6733 6733 Tomorrow_pos Correlation Coefficient .067** 1.000 Sig. (1-tailed) .000 . N 6733 6733 Spearman's rho Today_pos Correlation Coefficient 1.000 .101** Sig. (1-tailed) . .000 N 6733 6733 Tomorrow_pos Correlation Coefficient .101** 1.000 Sig. (1-tailed) .000 . N 6733 6733 **. Correlation is significant at the 0.01 level (1-tailed). I'm assuming 1-tailed because values are only positive. This comes as little surprise however because we would expect big moves to be followed by big moves. A technical ought to be available then (maybe Bollinger) which determines the size of moves to expect. Then rough trend to confirm this belief by linear regression is: Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) .009 .000 46.520 .000 Today_pos .210 .012 .210 17.586 .000 a. Dependent Variable: Tomorrow_pos Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) .009 .000 46.520 .000 Today_pos .210 .012 .210 17.586 .000 a. Dependent Variable: Tomorrow_pos So indeed the size of tomorrows price move is a positive function of todays price move (significantly). [Stats courtesy of SPSS.] Summary

to be summarised and expanded...

===

Adjusting the figures for inflation (which approximated to GNP/M4) the distribution of up/down was only slightly affected.

Yesterday was Up. Today was:

Up 26.9%

Down 25.0%

Yesterday was Down. Today was:

Up 25.0%

Down 23.0%

Total Up=52% versus down=48% so same as before.

Removing the upward bias hasn't changed many of the ups into down but it seems to have operated selectively. It has either turned UUs into UDs or DUs into DDs or taken UUs and made them DD (less likely). Suggesting that these were weak relationships.

For the BP data the effect is more obvious with only UU standing out as a feature of the data now.

UU 26.5%

UD 24.5%

DU 24.6%

DD 24.3%

Correlations
		VAR00001	VAR00002
VAR00001	Pearson Correlation	1	.210**
	Sig. (1-tailed)		.000
	N	6733	6733
VAR00002	Pearson Correlation	.210**	1
	Sig. (1-tailed)	.000
	N	6733	6733
**. Correlation is significant at the 0.01 level (1-tailed).

Correlations
			VAR00001	VAR00002
Kendall's tau_b	VAR00001	Correlation Coefficient	1.000	.067**
		Sig. (1-tailed)	.	.000
		N	6733	6733
	VAR00002	Correlation Coefficient	.067**	1.000
		Sig. (1-tailed)	.000	.
		N	6733	6733
Spearman's rho	VAR00001	Correlation Coefficient	1.000	.100**
		Sig. (1-tailed)	.	.000
		N	6733	6733
	VAR00002	Correlation Coefficient	.100**	1.000
		Sig. (1-tailed)	.000	.
		N	6733	6733
**. Correlation is significant at the 0.01 level (1-tailed).

OK the trends are still there after removing inflation.