Bulkowski’s Triangle Apex and Turning Points

Written by and copyright © 2008-2010 by Thomas N. Bulkowski. All rights reserved.

In the book, Elliott Wave Principle, authors Frost and Prechter write that the market often turns when price reaches the apex of a triangle. I tested this using three methods, and found it to be true in all cases. Price reaches a minor high or minor low 75% of the time within a few days of the triangle apex. Price turns from down to up or up to down 60% of the time.

Background

In the Elliott Wave Principle, Frost and Prechter write [page 51, second paragraph from page bottom], "...often the time at which the boundary lines of a contracting triangle reach an apex coincides with a turning point in the market." They are writing about the Dow Jones Industrial Average as a synonym for "market," and a "contracting triangle" is any of ascending, descending, or symmetrical triangles.

The following figure shows an example in a symmetrical triangle. Notice that point A in the figure, the triangle apex, lines up almost exactly with point B, a minor high, or "market turning point." Price turns at a minor high, changing the stock direction from up to down. Although the turn lasts only five days, price drops from a high of 22.52 to a low of 20.64, or over 8%.

Visual Test 1

I scanned through 388 stocks and found 221 ascending, descending, and symmetrical triangles. I counted how many of them showed the apex of the triangle meeting or coming close to (within a few days) a minor high or minor low. I found that 165 were close and 56 were not, for a success rate of 75%. The test was a rigorous one, meaning that I carefully drew the triangle trendlines and matched them up with the nearest minor high or low.

Visual Test 2

This test was less rigorous than the prior one. I let the computer draw the triangle boundaries and matched them up with price turning points. I was not looking for a minor high or low, but a change in price direction from up to down, or down to up, within a few days of the triangle’s apex. In the 239 triangles I looked at, 144 showed price changing direction and 95 did not, for a 60% success rate.

Both Tests 1 and 2 used my most recent database, covering 10/30/2006 to 1/27/2008 as the test period. The selected period allowed my software to display a complete chart, so the period was an arbitrary choice.

The following figure shows price nearing a minor high at B just as the apex of a symmetrical triangle forms at A. Price, however, does not change direction as the black arrow shows. Price falls before and after B.

Math Test 3

I used 500 stocks from July 1991 to July 1996 because the database contains the dates of the triangle apex for ascending, descending, and symmetrical triangles. I found all peaks and valleys within 31 days of each triangle’s apex using 5 days as the minimum peak or valley width. In short, I used a sliding window and found the highest high within that window and labeled it a peak, then I found the next peak and so on. I did the same for valleys. Then I found the peak and valley closest to the triangle apex and measured the distance.

I split the results from the three triangles -- ascending, descending, and symmetrical, -- and breakout direction, but the numbers are similar. Using the combined totals, I found that the average distance from the triangle’s apex to a minor high or low was 3.6 days.

Beginning with the 62 days window size (31 days to each side of the apex) and counting the number of minor highs and lows within that window and then using the distance from the first to last minor high or low, I found that the average distance between peaks or valleys was 13.1 days. In order for the apex to be closer than the average, it must be less than half this amount, or 6.55 days.

Think of it this way: imagine you are in a forest with trees spaced ten feet apart. If you stand between two trees, in the middle, five feet to your right would be a tree, and five feet to your left would be another tree. You represent the triangle apex and the trees represent the minor highs or lows. Stepping a foot or two in either direction places you closer to one tree than the average. That is what I saw using this test. The average distance from me to the nearest tree was 6.55 days, and yet I found that the apex was closer to one tree because it was just 3.6 days away.

In short, the apex is closer to a minor high or minor low than chance suggests.

-- Thomas Bulkowski