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Do stock prices reflect

The 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,

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) andВ  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).