The Chairman Victor Niederhoffer Home

9/25/2005
Polar Coordinates, by Victor Niederhoffer

A squall, a diversion, a relation, polar coordinates. The decline in stocks last week, the greatest in six months, led me to go beyond listening to the usual divertiment of the Schultz-Ebler transcription of the "Blue Danube Waltz" to the study of complex variables and polar coordinates. Polar coordinates are a way of representing a point in terms of its distance from the origin and the counterclockwise angle that the point makes with the x axis of the origin, an alternative to the usual graphing method. There are 2pi radians in a complete revolution. So 90 degree is 1/2 pi.

Radians are ideal for measuring the angles between two key variables such as bonds and stocks during big declines as well as other times. Some big down weeks in stocks, along with the change that week in bonds and the change in stocks the next week, follow.

With the kind assistance of Mr. Doc Castaldo, we have computed the angles measured in radians of all the concurrent moves in bonds and stocks, and tabled them along with the move in the S&P next week.

We studied all declines of 20 or more for stocks in a week.

Wk ending    SP move Bond move   Angle  Magnitude SP move next week
09/23/2005   -21.8    40.6       -0.50    46.1
04/15/2005   -40.0   221.9       -0.17   225.5     13.6
08/6/2004    -37.5   262.5       -0.14   265.2      2.6
04/30/2004   -33.4   -37.5       -2.30    50.2    -10.6
03/12/2004   -38.6    71.9       -0.50    81.6    -10.8
09/26/2003   -38.4   200.0       -0.19   203.7     34.0
03/28/2003   -30.3   203.1       -0.14   205.3     15.6
02/07/2003   -24.2    84.4       -0.28    87.8      6.5
01/24/2003   -42.8   131.3       -0.31   138.1    - 5.6
01/17/2003   -23.4   203.1       -0.12   204.4    -42.8
12/27/2002   -24.9   187.5       -0.14   189.1     38.1
12/13/2002   -26.5    81.3       -0.31    85.5     10.2
12/06/2002   -22.2   128.1       -0.17   130.0    -26.5
10/04/2002   -20.0    -9.4       -2.30    22.1     32.7
09/20/2002   -49.0    62.5       -0.67    79.4    -16.9
09/06/2002   -21.6    93.8       -0.22    96.3    - 3.4
08/30/2002   -24.9    93.8       -0.26    97.0    -21.6
07/19/2002   -73.3   - 9.4       -1.70    73.9      9.7
07/12/2002   -73.7   259.4       -0.28   269.7    -73.3
06/7/2002    -39.2   -34.4       -2.30    52.2    -19.4
05/24/2002   -22.8   134.4       -0.17   136.3    -14.9
04/26/2002   -54.7   165.6       -0.31   174.4      1.0
04/5/2002    -24.2   246.9       -0.10   248.1    -13.4
02/08/2002   -26.9    53.1       -0.47    59.5      8.3
01/11/2002   -27.5   325.0       -0.08   326.2    -18.7

Angles in radians, measured from bond axis (x-axis)

Comments:

• Bond moves in these weeks are enormous and mostly positive
• All stock moves considered are negative, hence all angles are in third or fourth quadrant (negative angles)
• The convention that Bonds are plotted on x-axis and Stocks on y-axis is based on EdSpec p. 287 (Weaver Charts)
• Weeks end on Friday

One finds no systematic relations between the radian measure of the bond stock move in the big stock decline weeks, and the subsequent week during the previous three years. However, there is a clear tendency for the lower magnitudes of stock bond moves to be associated with rises in stocks the next week. It is interesting to note in this regard that the magnitude of the bond stock move in the week ending 9/23 was the smallest of all 25 considered.

Rational Trig and Markets, by Doc Alex Castaldo

Last week the Bloomberg newswire carried the surprising announcement that Australian mathematician Norman Wildberger has developed a new form of trigonometry that does not use sines, cosines, etc., but can nevertheless solve all trigonometric problems (and do it better, he claims, than the classical trigonometry invented 3,000 years ago). Called Rational Trigonometry, Wildberger's invention does not even use distance and angle, but replaces them with two new notions: "quadrance" and "spread" (see below).

I was skeptical, to say the least, but after reading the first chapter of his book, I see nothing obviously wrong or nonsensical in what he is saying. Here is a brief summary:

Def. The "quadrance" between two points is the square of the distance, i.e. the quadrance between A_1 = [x_1, y_1] and A_2 = [x_2, y_2] is defined as

Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2

Def. The "spread" betwen two lines L_1 (with equation a_1*x + b_1*y + c_1 = 0) and L_2 (with equation a_2*x + b_2*y + c_2 = 0) is defined as

s(L_1, L_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 + b_1^2)*(a_2^2 + b_2^2))

The spread is 0 when the lines are parallel, is equal to 1 when the two lines are perpendicular. [The spread is like a sine squared, but remember that we are not supposed to use sines and cosines anymore].

Wildberger claims that all trigonometric problems can be solved by applying what he calls The Five Main Laws of Rational Trigonometry.

We are given three distinct points A_1, A_2 and A_3. We use the notation Q_1 = Q(A_2, A_3) etc. and s_1 = s(A_1 A_2, A_1 A_3) etc.

1. The Triple Quad Formula
   The three points are collinear iff (Q_1 + Q_2 + Q_3)^2 = 2*(Q_1^2 + Q_2^2 + Q_3^2)
2. Pythagoras Theorem
   The lines A_1 A_3 and A_2 A_3 are perpendicular iff Q_1 + Q_2 = Q_3 [this is familiar]
3. The Spread Law
   For any triangle with non-zero quadrances s_1/Q_1 = s_2/Q_2 = s_3/Q_3 [This is recognizable as similar to the Sine Law, but with all terms squared]
4. The Cross Law
   For any triangle (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 * Q_2 * (1-s_3)
5. The Triple Spread formula
   For any triangle (s_1 + s_2 + s_3)^2 = 2*(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3

Does it really work? Does it offer any advantage over classical trigonometry? Even if it does, will people want to abandon familiar concepts to learn the new ones? I doubt it. But Manchester Trading will try to be the first to apply these concepts to financial markets.

Charles Sorkin comments:

Do you really want a world with "radial technical analysts?"

Actually, the development of price patterns generated radially throughout a particular time period could allow visualizations of patterns that would never be noticed on Cartesian coordinates. It could work spherically as well. Consider a representation of intraday security prices represented parametrically with a reference to time. The chart would start in the center with time=0. Each new tick could be described with a price and with accumulated volume (or perhaps on-balance-volume). Radius would expand at a constant rate throughout the day. One would need to develop a simple function to convert those variables into radians.

Others are also looking at new graphical representations of prices, I see. Bloomberg for instance has a new display that looks for (abnormal?) skewness in a chain of option prices.

Kim Zussman comments:

One aspect of this would be to think of the unit circle with vector radius. At time zero, the vector points right and angle is zero. After time t, SPY and TNX (10 yr yield) have moved, and the difference between their moves determines the angle the vector makes with the X-axis. The conversion from % difference to angle units would seem arbitrary, but maybe there is something to thinking about divergence (convergence) between stock and bond returns as an angle with predictive value over many occurrences.

I looked at SPY, and this week's drop was not as big in % as many others mentioned (I am not sure why you look at points but assume this scales because you test futures). However for 2-weeks, the decline was over 2%. And interestingly, TNX (which moves inversely to bond price) went up in recent 2 and 3-weeks.

Looking at weekly returns for SPY and TNX since 1/96, week's return after SPY lost>2% the prior 2-weeks, and TNX was higher than 3-weeks ago:

AVG     1.009827998 (+1%)
     SDEV    0.026421589
     COUNT            33
     Z       2.136796096

Of course none of this takes into account "weather" effects such as Fed week tightening (Mr Cohn got it wrong this week) and unusual hurricane frequency in the face of rising oil prices.