PTR's Fibonacci Trend Theory(tm)
The Fibonacci Summation Series produces a exponential growth curve that is nearly the same as the exponential growth curves generated by those investments that earn and compound interest or capital gains. Since the stock market is essentially based on the mathematics of compounding interest, the vast majority of long term price charts will have "a very strong tendency" to follow upward sloping straight lines "in logarithmic" chart scale, or log-linear (semi-log) chart scale. That is because exponential curves appear as steeply rising curves in linear scale, but as straight lines in semi-log scale.
In the late nineteenth and early twentieth century, calculating an exponential curve would have been a daunting tasks, so we believer that speculators either began using the simpler Fibonacci Series as an "approximate" proxy for the exponential curve, or they just inadvertently, and incorrectly, linked the significance of the Fibonacci series to something that was actually being produced or influenced by the mathematics of compounding, and had no real connection to the Fibonacci series. Never the less, regardless of how these numbers came to be tied to stock prices and price-time patterns, it is very clear that they are currently connected with a high degree of predictability.
The introduction to this section explains the process for "locating" one or more "possible" existing AND future "mean" trend lines that price "may have" a tendency to follow more then a random and unpredictable path. The long term exponential regression "mean" tells you where the price average is now and where it has been in the past, but the Fibonacci trend seeks to determine a "mean" line that price is "somewhat likely, or "very likely," to be following in the future.
Copyright(c) 2003
The PriceTime Review
Andrew J. Quiggly
Editor
S&P 500 "cash" Index (SPX) 5/25/04.
One of the major tenets of PTR's Fibonacci Trend Theory(tm) is that stock prices, or their indexs, will have a "very good" to strong tendency to form straight trend and mean lines in logarithmic chart scale, or semi-log scale, and these trends will also have a "good" to "very good" tendency to become continuous from any major high to the next major low, or from the prior major low to the next major high. By experimenting with some possible trends that meet the "theory" requirements, we can "many times" identify one or more "possible" MEAN trend lines that "may" aid us in forming our current opinion and future expectations.
The chart at the bottom of this page deals with the long term trends, and the page at this link <SPX: NT Fibo. 5/25/04> considers only the near term "expectations" for SPX. Needless to say, the two are connected at the hip, and I urge everyone to read both if they have the time.
The long term chart of the SPX index, below, shows an excellent example of another important tenet (doctrine) of our Fibonacci Trend Theory. When placing a Fibonacci trend we want to "locate" a trendline that touches as many of the points where actual price crosses over, touches, or bounces off a Fibonacci number line and thereby keep the "open gaps" to a minimum.
An "open gap" is those places where actual price "did not" touch the "hypothetical" Fibonacci trendline and the Fibonacci number line at the same time. This "gap" is illustrated on the top chart by the area between Fibonacci number lines 144 and 233, during 1975 to 1980. If we are "assuming" that the actual regression mean line is our "hypothetical" Fibonacci trend line, then we can see that we have two large open areas or "gaps" during 1940-1960, and then again from 1965 to 1985, where price (index value) "does not" touch our mean trendline AND the Fibonacci number lines "at the same time."
Now, a "perfect" Fibonacci trendline should not have any "Fibonacci gaps;" that is to say, there "should be" a straight line, in semi-log scale, that connects the lowest "average" price to the highest "average" price that passes through, or otherwise touches, very Fibonacci line between that high and low, AND the "time periods" that elapses between each "crossing" (step) are equal or close to being equal. Now, if, and only if, the time "steps" are the same, and the price passes through the Fibonacci line numbers at those time "steps," do we have a CONTINUOUS MATHEMATICAL FUNCTION that is both an "exponential growth function" and a Fibonacci summation series that is also a exponential function.
The Fibonacci Summation Series produces a exponential growth curve that is nearly the same as the exponential growth curves generated by those investments that earn and compound interest or capital gains. Since the stock market is essentially based on the mathematics of compounding interest, the vast majority of long term price charts will have "a very strong tendency" to follow upward sloping straight lines "in logarithmic" chart scale, or log-linear (semi-log) chart scale. That is because exponential curves appear as steeply rising curves in linear scale, but as straight lines in semi-log scale.
In the late nineteenth and early twentieth century, calculating an exponential curve would have been a daunting tasks, so we believer that speculators either began using the simpler Fibonacci Series as an "approximate" proxy for the exponential curve, or they just inadvertently, and incorrectly, linked the significance of the Fibonacci series to something that was actually being produced or influenced by the mathematics of compounding, and had no real connection to the Fibonacci series. Never the less, regardless of how these numbers came to be tied to stock prices and price-time patterns, it is very clear that they are currently connected with a high degree of predictability.
The introduction to this section explains the process for "locating" one or more "possible" existing AND future "mean" trend lines that price "may have" a tendency to follow more then a random and unpredictable path. The long term exponential regression "mean" tells you where the price average is now and where it has been in the past, but the Fibonacci trend seeks to determine a "mean" line that price is "somewhat likely, or "very likely," to be following in the future.
Copyright(c) 2003
The PriceTime Review
Andrew J. Quiggly
Editor
S&P 500 "cash" Index (SPX) 5/25/04.
One of the major tenets of PTR's Fibonacci Trend Theory(tm) is that stock prices, or their indexs, will have a "very good" to strong tendency to form straight trend and mean lines in logarithmic chart scale, or semi-log scale, and these trends will also have a "good" to "very good" tendency to become continuous from any major high to the next major low, or from the prior major low to the next major high. By experimenting with some possible trends that meet the "theory" requirements, we can "many times" identify one or more "possible" MEAN trend lines that "may" aid us in forming our current opinion and future expectations.
The chart at the bottom of this page deals with the long term trends, and the page at this link <SPX: NT Fibo. 5/25/04> considers only the near term "expectations" for SPX. Needless to say, the two are connected at the hip, and I urge everyone to read both if they have the time.
The long term chart of the SPX index, below, shows an excellent example of another important tenet (doctrine) of our Fibonacci Trend Theory. When placing a Fibonacci trend we want to "locate" a trendline that touches as many of the points where actual price crosses over, touches, or bounces off a Fibonacci number line and thereby keep the "open gaps" to a minimum.
An "open gap" is those places where actual price "did not" touch the "hypothetical" Fibonacci trendline and the Fibonacci number line at the same time. This "gap" is illustrated on the top chart by the area between Fibonacci number lines 144 and 233, during 1975 to 1980. If we are "assuming" that the actual regression mean line is our "hypothetical" Fibonacci trend line, then we can see that we have two large open areas or "gaps" during 1940-1960, and then again from 1965 to 1985, where price (index value) "does not" touch our mean trendline AND the Fibonacci number lines "at the same time."
Now, a "perfect" Fibonacci trendline should not have any "Fibonacci gaps;" that is to say, there "should be" a straight line, in semi-log scale, that connects the lowest "average" price to the highest "average" price that passes through, or otherwise touches, very Fibonacci line between that high and low, AND the "time periods" that elapses between each "crossing" (step) are equal or close to being equal. Now, if, and only if, the time "steps" are the same, and the price passes through the Fibonacci line numbers at those time "steps," do we have a CONTINUOUS MATHEMATICAL FUNCTION that is both an "exponential growth function" and a Fibonacci summation series that is also a exponential function.
Unfortunately, the real world is not as nice and clean, or accurate, as the hypothetical basis of our theory, so in actuality we have to find the trend that best meets the most criteria while not expecting perfect patterns. The underlying mathematical basis of the Fibonacci Trend Theory(tm) is the exponential growth function, and the underlying basis of this exponential growth function is the compounding interest equation.
However, we cannot expect a stock or index to "always" follow it's "hypothetical" long term trend, and it would be totally erroneous to assume that price (index value) "must always" touch that trend at the time when it crosses the Fibonacci number lines. The reality is, that this does happen so often that we, at PTR anyway, have come to expect it; eventhough, we know better then most that you have to "assume" some "range" or tolerance when dealing with this form of analysis. Now, if you look at the lower chart, at the bottom of this page, you can see that we found (placed) two more "hypothetical" trendlines that "fit" better than the regression mean, as shown by the smaller area where there is (was) an open "gap,"
As best I can tell, what is happening here is that we are getting just the tops and bottoms of the "spur" touching the longer term trend, and that mean is "fairly" close to the one that will be valid for many years to come. Also, while it "was" bullish that price reversed of this line at the October, 2002, low, I wouldn't expect that line to be anything more than major support in the future, and I would eventually expect it to signal a "crash" if it's violated.
While it is true that price "should" eventually "regress to the mean," it is equally true that more often than not we see excessive upside spur trends, that get well above the long term mean, balanced out by excessive negative spur trends below that mean. Of course, I could be totally wrong about this, and that "mean" line will hold up price for many more years to com. To addressee that statement, all I can say is that I do not have any crystal ball that will actually predict the future, and I firmly believe that the future IS NOT preordained to some exact specification, or if it is, then GOD is the only one who knows what it will be, and he is not confiding me...as far as I know!
Before I leave this analysis, I want to point out a few historic facts about the "deka (deca-) boundary lines" at 10, 100, and 1000. As you can see, on the top chart, the SPX price (index value) was "hung up" (consolidating) in the area around these lines for many years after it initially reached them in a bull leg up. The index was "hung up" around 10 from 1922 to 1942 (20 years), around 100 from 1966 to 1982 (16 years), and now it has only been "hung up" around 1000 from 1998 to 2004 (six years).
While anything is possible, and no where is it chiseled in stone that stock market history "must" repeat itself, I "highly suspect" that this is another piece of circumstantial evidence that supports the bearish point of view. On the flip side of this thesis, who is to say that another run up to a double top, or even new highs, can't happen like it did in 1929, with that deka boundary then pulling the index back down in another super crash.
While I consider that "very unlikely," it's hard to say what a fresh batch of billions (or many billions) can buy before those foreigners who are trading their life savings for U.S. IOU's finally figure out that: Hey Yank, I'm losing serious money here!
For the PriceTime Review
Andrew J. Quiggly
Editor <SPX: near term Fibo. 5/25/04>
