parabolaAn interesting and scholarly article on basketball angles and spins in the Washington Post, kindly sent my way by Dan Grossman of Florida, indicates that physics calculations show that the angle with which a shot is thrown influences its accuracy. The gist is that the higher the arc, the greater then chances it will go through. Also, the higher the height the ball is launched from, the greater the margin of error on the shot, and the greater the success. Taking one thing with another, a launch angle of between 45 degrees and 55 degrees seems best.

Angles have played a big role in stock market technical analysis. The most influential angles are those derived from the work of W. D. Gann who believed that when price breaks through an angle of 30, 45 or 60 degrees from a previous low to previous high, a continuation in direction of the break through was possible. As to how to compute the previous low or high, where it should start, where it should end, and what the time scale should be — why, that's why there are 1.4 milion entries on the subject of Gann angles with most concluding that they have been superceded by more sophisticated tools in these days of modern computers.

Being one to believe that taking the pencil to paper might help to find regularities, I computed some subsequent prices moves for the last 15 years for daily, weekly, and monthly prices. I looked at the angle between the two last moves of an index. For monthly prices, I looked at the subsequent moves following the moves in the previous two months. For example I looked at the moves in January and February to predict March. For weekly moves, I looked at the two previous weeks to predict the third week. For exampe the first and second week of the month to predict the third. For daily moves, I looked at the previous two days to predict the third. For example, Monday and Tuesday to predict Wednesday. The angles formed could be computed by using trigometric identities to compute the adjacent angles and noting that the sum of the three angles is 180 degrees. Here are the results:               

subsequent period   prior     previous    expectations   number of obs

monthly                    +         +                       4                53

monthly                    +         -                      -4                 41

monthly                     -         +                      7                 40

monthly                     -          -                      -6                33                                                                                                              

weekly                      +         +                      -2               185

weekly                      +         -                       4                186

weekly                       -         +                      -3               189

weekly                       -         -                        1               147          

daily                        +         +                      -0.7             952

daily                        +         -                       -0.5             898

daily                        -         +                        0                904

daily                         -       -                          1    744                                                                                                          

The many regularities that this table reveals are a good start to find angles that are useful for success in shooting for profits in markets. 

Bill Rafter offers:

If there is some mystical or otherwise relationship of price to time, it would perforce need to dictate the scale of the price charts for any angles to describe that relationship. In earlier times graphs and charts were drawn in pencil on standard graph paper, such as 10 x 10 to the inch. If everyone kept the same charts in the same scale, an argument can be made (i.e. the self-fulfilling prophesy) for various angles, Gann or otherwise. However the person who used a scale not similar (in the geometric sense of the word) to the standard scale would have different points of intersection with respect to the same angles as the standard scale. Fast-forward to today and we find computers rescaling charts to fit on the visual page to the extent that no longer is there a standard scale. So at the very least, there can no longer be a self-fulfilling prophesy. Whether there is a price-time relationship, I do not know. But the next time I see Master Yoda I will ask him.

Dr. Rafter is President of Mathematical Investment Decisions, a quantitative research consultancy

Jim Sogi writes:

 One of the problems is asymmetry. Contrary to symmetry theory, the markets are not symmetrical. It makes the Gann ideas or other symmetrical equations inaccurate. This includes normal distributions.

Sushil Kedia comments:

A few years ago Mr. Sogi had written a review of a book from a publisher in Australia providing co-ordinate geometry treatments of the Mathematics of the Divine Ratio. The Chair and Dr. Castaldo too have written about the applications of polar co-ordinates to transforming prices to study appropriately, thereon. Perhaps a number of ideas from that book could be applied to produce suitable transforms and then fitments to the effect of say treating the rate of drift as equivalent to an angle of zero or one would provide perhaps some worthy examinations of the other standard angles as 1/3rd or 2/3rds (pi/12 or pi/6) etc.

Jim Sogi replies:

Why is it that the last few weeks, the price followed an upper and lower line so well as moving mechanically upward? It has kind of broken that recently, but it seems more than random things the mind puts together like a taking patterns out of random dots. I've graphed random series and seen the so called trends, but they don't seem as linear. And it seems to continue for days and weeks.

On the subject of the negative time idea Rocky talked about earlier, and since it's spring now, a look at our Gregorian calendar shows the inaccuracies in the counting system. Assume as a mind experiment that the market operates on absolute time. Given an inaccurate Gregorian system, at times the measurement of absolute time might appear to jump or reverse at various adjustment points. See the wikipedia entry on Julian Day Calculation and on the accuracy of Gregorian Calendars.

Pitt T. Maner III writes:

John WoodenThe discussion on shooting angles in basketball reminded me of one of John Wooden's favorite ideas which is to use the backboard or "glass".

Within days of implementation of this new 'back' board, players found that they could gauge the angle and use this new tool to their advantage to make shots. The problem today is that players have either forgotten or never learned its full advantage. Arguably the most successful coach of his era and probably of all time, UCLA Hall of Fame Coach John Wooden was renowned for teaching his players to use the glass both in the post and especially off of fast break transition. Coach Wooden's teams won 9 straight NCAA Titles (10 in 11 years) a feat unparallel before or since in collegiate basketball. Coach Wooden believed that if his players practiced making 5-10 foot bank shots at the 45 degree angle off the break and could make them at a rate of about 75-80% these shots were nearly as good as getting a lay-up. For those young players that missed the opportunity to see the great UCLA teams they were almost always successful in transition. Probably the single greatest scoring performance ever in an NCAA Championship Game was by UCLA's Bill Walton’s 21 for 22 field goal shooting performance against Memphis in the 1973 title game. Bill Walton’s' patented move was the up and under using the glass.

I have found that it is possible to use the target square on the backboard to advantage when shooting free throws and banking shots in instead of trying to get a "swoosh" each time. Having a visual target behind the "bucket" seems to help.

Banking in shots, however, is probably best used (like Wooden suggests) when shooting from the sides in basketball. Wooden believed in playing percentages and when a basketball hits the backboard it has almost a "second" chance of going in as opposed to hitting the front or side of the rim or completely overshooting or undershooting the mark. I think the coach would have preferred a sure layup over a dunk any day—and how many times has one seen a sure dunk pop out when thrown down with too much force when a layup would have scored two.

Not quite sure what a market analogy would be for a backboard–perhaps some type of straddle where you can make money on the continuation of the angle in either direction?At any rate one of the fun aspects of the NCAA basketball tournament is getting a chance to see various styles of play and the leveraging of seemingly higher risk approaches. The Cornell–Kentucky game on Thursday night, for instance, if it stays close, could be an example of excellent, confident outside shooting and defense vs. more pure interior power, younger talent and athleticism (and good outside shooting too). But as they say, you can "live and die by the 3-pt. shot".


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