# Angles, from Bruno Ombreux

September 25, 2006 |

There are many angles to the markets.

There are Gann angles, which of course make no sense, because angles on a chart are depending on axis units.

There is also the Cauchy distribution, whose fat tails are scaring away those who can remember 1987 but have never traded smallcaps or electricity. This distribution can be generated from angles, made by someone shooting randomly at a distant target. Where "randomly" means uniform. Hence fat tails are the property of some distribution of angles.

More interestingly, there seem to be a lot of statistical tests developed for circular data; that is angles. I found out about them in 100 statistical tests by Gopal K. Kanji. Just for fun, I gave a try to the V-test, or Modified Rayleigh. It is a test for randomness, checking whether observed angles tend to cluster around a given angle.

The data is JPY/USD monthly returns since 1965. One problem surfaced though — how to transform returns into angles?

I chose to project them on a vertical axis, in some reminiscence of the Cauchy target experiment. As a result, time is factored out of the study. It could turn it into what I think they call an "axial study", in which all angles need to be doubled in the computations. This could be a mistake, but it does not affect the results. The conclusion is the same whether the angles are doubled or not.

And the conclusion is that we reject the null hypothesis that angles are random around the zero-line, at the 0.0001 significance level. Rayleigh's V is 5.255. There is some element of non-randomness in monthly JPY/USD, which still need to be identified.

Here is the code for the test, except that V's significance had to be checked in a table, unavailable in any R package.

Close <- 100*diff(log(YEN\$Last))
Angle <- 2*atan(Close) #Axial data

xbar <- sum(cos(Angle))/length(Angle)
ybar <- sum(sin(Angle))/length(Angle)
r <- sqrt(xbar*xbar+ybar*ybar)
theta0 <-0 # theoretical angle direction i.e. null hypothesis
nu <- r*cos(phi-theta0)
V <- nu*sqrt(2*length(Angle)) I am no Gann fan, but Gann angles escape the aspect ratio problem you mention because Gann plotted his charts on paper of a constant scale, so that one increment of time was always kept constant in physical distance relative to the units of price on the y-axis. Typically it was 1 point in price = 1 unit of time. In this way, a 45 degree line always related to a rate of ascent or descent of, say, one point per day. So there was some internal consistency. Or, as Shakespeare put it "Though this be madness, yet there is method in't."

Early technical analysis software packages like CompuTrac let you draw angled lines which were completely arbitrary because of the vertical range in the time period plotted in the chart, but they were there because the TAG group threw just about anything into the program that users asked for. But they also had methods of drawing Gann angle lines with the consistent aspect ratio ability.

Bruno replies:

To answer my own post, I tried as much as I could to find randomness with circular tests, but could not. The reason is certainly that returns data is not circular. The way to make it circular is to introduce time. Divide the circle in 12 for monthly returns.

Then returns have to be in a third dimension. Fortunately, we have got spherical statistics! With all due respect, Gann realized his error only after he had been publishing for some time. It was a retrospective fix. But it is an inadequate fix.

When a stock splits 2:1 any normal chart is now no longer a 1 point to 1 time unit ratio. When Microsoft paid its 10% dividend is the new ratio now .90 to 1.00 or should it be 1.10 to 1. Gann is moot on the question.

When a stock pays dividends should the price be adjusted? Should the dividends just be ignored with the attendant error in rate of return?

What about weekly charts - is the time scale 5 units or 7 or just 1 (a week)?

What about monthly? Is it 1 unit, 21 units or 30 or the actual number of trading or calendar days?

With many angles, many time scales and dubious rules it should be quite easy to find many examples that come 'close' to turning points in the markets. In fact it is probably quite difficult to find any failures. This is especially true if one is allowed to define 'close' as whatever one needs to make the current data fit.

Even though the arcane mysticism of Gann is suspect, Bruno's ideas on angles may have some merit and should not be lumped into the same basket. The Cauchy distribution induced by the angle model can be problematic. During the 90's several advances were made by Zar and others in the statistics of angles and tests thereon. That area is relatively new but very workable.