Do stock prices reflect Fibonacci ratios? PDF
Preview Do stock prices reflect Fibonacci ratios?
Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel T he claim has been frequently made that Fibonacci ratios occur in stock market data. A Fibonacci ratio is the ratio between any successive numbers of the Fibonacci sequence — the Fibonacci sequence 1,1,2,3,5,8... produces the ratios 1, 1/2, 2/3, 3/5, 5/8.... After the first four numbers in the Fibonacci sequence, the ratios approximately equal 0.618, known as the Golden Ratio or phi (f ). Comparing one number in the sequence to the next lower number (8/5,5/3, 2/1. . .) produces ratios that approximate 1.618, the inverse of the Golden Ratio, f In their book, Elliott Wave Principle, Key to Stock Market Timing, A. J. Frost and Robert Prechter cited Dow theorist Robert Rhea's study of nine Dow Theory bull markets and nine bear markets . Of the 13,115 calendar days reviewed, bull markets were in progress for 8,143 days and bear markets for 4,972 days, giving a ratio of 0.611, a value close to the Golden Ratio, f = 0.618. In another Rhea study, the sum of the "primary swing" advances during a particular bull market divided by the advance of the bull market was 1.621, a value close to 1/f . I designed a study to test for Fibonacci ratios in stock market data. Rhea, himself, concluded in his book, The Story of the Averages, that the "figures show that bear markets bear no particular relationship to the preceding bull periods so far as percent of retracement is concerned." Furthermore, the corresponding figures for percent of net advance in primary swings during the other eight bull markets are not as close to the Golden Ratio as the one cited by Frost and Prechter. Elliott Wave Theory The Elliott Wave Theory of the stock market is based on the assumption that stock prices as a whole move up in five waves and down in three during a bull market (the opposite is true in a bear market) and Article Text Copyright (c) Technical Analysis Inc. 1 Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel that every wave is, itself, subdivided in the same way (Figure 1). Such a chart of stock prices would appear similar on both a small and large scale (termed "self-similar"). While the Elliott Wave Theory has a large following, there is no scientific model explaining this behavior, and many practitioners of stock market technical analysis are skeptical of the theory, believing that anything can be "read into" the stock charts. In academic circles, it is the random walk theory of the stock market that is generally accepted. Random walk says there is no sequential correlation between prices from one day to the next, that prices will act unpredictably as they seek a level in response to supply and demand. Because Elliott waves are constantly subdivided, a ratio approaching the Golden Ratio, f = 0.618, might be anticipated. For similar reasons, ratios of total advances over total declines and average advances over average declines in a given wave might be expected to approach the Golden Ratio. I also investigated the ratio of total advances over new advances (or declines in the case of a down wave). Methodology The main problem in attempting an objective study of stock prices is the identification of trends. What seems like a downtrend to one observer is just a minor correction in an overall uptrend to another. As an objective criterion, I used moving average crossovers. If the current value of a stock index is larger than the moving average, the trend is up (and vice versa). Points where the trend changes are called crossovers. An upwave is the interval from the minimum that lies between a negative and a positive crossover to the maximum that occurs before the next crossover. A downwave is defined correspondingly (Figure 2). I used stock price data for the Dow Jones Industrial Average from May 13, 15185 to January 30, 1986 because it contained one of the most distinctly pronounced advances in recent years (from September 18, 1985 to January 7, 1986) and any hypothesis developed from this data could be tested on more recent data. My program determined the upwaves and downwaves and, for each wave, calculated the number of advancing days, the number of declining days, total amount of advances, total amount of declines and, where these values were not zero, the average advance, the average decline, number of advancing over declining days, total advances over declines, average advances over declines, and total advance (decline) over new advance (decline). I ran the program for moving average cycle lengths of 5,10, 15,20, 25, 30,35,40, 45,50, 60,70, 80, 90 and 100 days. In addition, I manually compared the lengths of subwaves in the upwave from September 18, 1985 to January 7, 1986 and determined the ratios of the total number of advancing days over declining days in this wave. I examined the neutral period of June 20,1985 to September 18,1985 using the same ratios. Figure 3 are observations for the 10-,20- and 30-day moving averages. Moving averages of cycle lengths from 5 to 100 days gave similar results. Assuming the market behaves similarly whether the trend is up or down, I combined observations for up- and downwaves, using reciprocal ratios for downwaves (declining ÷ advancing days instead of advancing ÷ declining days). There is no obvious consistency in these values. I did a two-tailed t-test on each of the ratios to statistically test the null-hypothesis that the mean value of the ratio is 1.618. The null hypothesis says there is no validity to the claim that two variations of the same thing can be Article Text Copyright (c) Technical Analysis Inc. 2 Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel FIGURE 1: Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel FIGURE 2: Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel FIGURE 3: NA = number of advancing days, ND = number of declining days, A = total advances, D = total declines, A = average advance, D = average decline, D= net advance [decline]. Copyright (c) Technical Analysis Inc. Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel distinguished by a specific procedure. In hypothesis testing, the null-hypothesis is never accepted, but can be rejected if warranted by the data. A small level of significance is desirable because it represents the probability of mistakenly rejecting the null-hypothesis when the hypothesis is, indeed, true. Hence, rejecting the null-hypothesis in these tests means that, with only a slight probability of error, the tested ratio is not 1.618. For the ratio of total advances over net advances, the null-hypothesis could be rejected at the 1% level of significance for 10-day moving averages. For 20- and 30-day moving averages, the null-hypothesis could not be rejected at this significance level, but it could be rejected at the 10% level for 20-day moving averages. For the ratio of advancing days to declining days, the null-hypothesis could be rejected at the 1% level for 10-day moving averages, but could not be rejected for the 20- or 30 day moving averages. For total advances vs. total declines, the null-hypothesis could be rejected at the 5% level for 10-day moving averages, but could not be rejected for the 20- or 30-day averages. Finally, for the average advance vs. the average decline, the null-hypothesis could be rejected at the 10% level for 10-day moving averages, but could not be rejected for the 20- or 30 day moving averages. Subwaves of an impulse wave I opportunistically selected the advance from September 18, 1985 to January 7, 1986 for its distinctive shape. Within this wave, a 10-day moving average distinguished nine subwaves with the following advances or declines: The resulting advance/decline ratios of upwaves to subsequent downwaves were 2.534,2.982,5.903 and 2.758, respectively. The number of advancing days from September 18, 1985 to January 7,1986 is 50 and the number of declining days is 26, giving a ratio of 1.923. The ratio of total advances to net advances is 1.338. From these, I can deduce no connection to Fibonacci numbers or the Golden Ratio. The period from June 20,1985 to September 18,1985 also was selected opportunistically as a typical neutral period consisting of one upwave and one downwave. During this period there were 35 advancing and 26 declining days, giving a ratio of 1.346. The average advance was 4.606 points and the average decline 6.108, giving a ratio of 0.754. Article Text Copyright (c) Technical Analysis Inc. 3 Stocks & Commodities V. 7:12 (433-436): Do stock prices reflect Fibonacci ratios? by Herbert H.J. Riedel What seems like a downtrend to one observer is just a minor correction in an overall uptrend to another. Conclusion In my findings, none of the ratios examined for the waves defined by moving average crossovers have a mean of 1.618. This was clearly shown for waves defined by 10-day moving averages. While the data does not allow me to reject the null-hypothesis that the ratios are equal to 1.618 for waves defined by 20- or 30-day moving averages (with one exception), this does not mean the null-hypotheses are accepted. Rather, the assumed "self-similarity" between waves of different degrees suggests that the same results should be obtained regardless of the degree of the waves. Therefore, the inability to reject the hypothesis is most likely due to the small sample size. In any case, all the ratios, for all moving average cycle lengths, show a large measure of dispersion with standard deviations on the order of the respective means, suggesting that the cycle lengths are poorly estimated. There are many reasons why a Fibonacci relationship might not show up in an investigation of this kind, even if some such relationship exists. For example, the method of determining up- and downwaves using moving averages, while objective, does not allow for any "internal" wave structure, such as an "irregular correction" to an upwave, which might extend to a new high, although it properly should be counted as a downwave. Or, possibly, I looked at the wrong ratios, or perhaps the 182 days of data were insufficient for any kind of consistency to become apparent. Nevertheless, I believe that if there were an underlying relationship between Fibonacci ratios and stock prices, some evidence would have turned up in the data studies I performed. And so, I believe the reason we find a great variation in advance/decline ratios rather than some consistent ratio is that stock market trends can occur in any degree of intensity or steepness, and this would naturally be reflected in the ratios that were examined. Herbert Riedel, Ph.D., Box 93, The Citadel, Charleston, SC 29409, is an assistant professor in the Department of Mathematics and Computer Science at The Citadel, the Military College of South Carolina. His doctorate is in pure mathematics from the University of Waterloo, Ontario, Canada (1984) and he has traded stocks, options and futures since 1982. Figures Copyright (c) Technical Analysis Inc. 4
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