In the previous article (publication 02-03) we considered nonrandom walk of the Dow Jones Industrial Average and showed that observed nonrandom effects can be explained by the positive correlation of the price changes. In other words if one observes a trend then there is a relatively high probability of the trend continuation.

Here we will continue this study trying to understand price trend stability more deeply. As in the previous publication we consider the historical prices of the Dow Jones Industrial Average for the period from 1900 to 2001.

Definitions
p(i) closing price at trading day # i

Delta p(j) = [ p(i+j) - p(i) ] / p(i) * 100%       relative price change

y(t) = A + B * t      equation of linear fitting the closing prices p(i), p(i-1), ..., p(i-N+1)

N is the number of fitting points

b = B / (A + B * i) *100%     normalized slope of the linear fitting. It is equal to the average   % price change per day.

The trend is positive if B (or b) > 0. The trend is negative if B (or b) < 0.

Problem 1
Suppose one observes a positive price N day trend. It means that fitting line over N trading days has a positive slope (B > 0). What is the probability of positive price change after j days?

We considered j = 1, 2, ..., 10. It is from 1 to 10 day  price change. The results are shown in the figure. Black squares are the average results for any slope. The average probability of growth is greater than 0.5 because of positive long term trend of the Dow. One can see small growth of the probability when j becomes larger.

The red and green points shown the probabilities of growth after short trends (5 day prices were used for linear fitting) with positive (green) and negative (red) trends. One can see that the probability of growth is about 1.5% greater after positive trends and 1.5% smaller after negative trends.

Calculations for larger N (10, 20, and 40 days) showed that probability of growth change slightly from the average value. One can conclude that for the Dow short trends have larger probability to be continued.  More details will be considered in the next section.

Problem 2
It is more important for investors and traders to study the % price change. It is the real money, not math tricks with the probability. We calculated the price change for j = 1, 2, ..., 10 days for N = 5, 10, 20, and 40 days. Therefore we considered short and intermediate trends of the Dow. The results of calculations are shown in the figure. We showed normalized price changes for various j. Normalization has been done by dividing price change in % by the average price change for the corresponding j.

One can see that the largest effects of the trend are observed for N = 5 for negative and positive slopes. It is consistent with the calculations of the probabilities of price growth. It is very interesting to note that the largest price change is observed for small j. Therefore, short-term traders can possibly benefited form using short term-trends for making trading decisions.

However the life is not simple. The risk of such sort-term trading is relatively high. The next figure shows the risk/return ratios for different j for N = 5 and b > 0. (please read more about risk/return ratios in our handbook).  One can see that' the risk/return ratios are very high for small j and such short-term trading can very risky.