Unexpectedly Intriguing!
28 January 2009

Eskimo Hat Galaxy (Strange Attractor model) In our fourth anniversary post, we demonstrated that for much of the current century, changes in the growth rate of stock prices have closely anticipated changes in the growth rate of their underlying dividends per share.

In doing that, we showed that given the circumstances that have prevailed in the stock market since 2001, if one were equipped with a reliable picture of the future of stock dividends per share, one could work out how much stock prices would change. That turns out to be driven by a very simple mathematical relationship, very similar to Newton's Second Law: F = ma. For our case, F would be replaced by the change in the growth rate of stock prices (Ap), m (for our purposes, an amplification factor) would appear to be a constant and the variable a would be replaced by the change in the growth rate of dividends per share, or perhaps more appropriately, the acceleration of dividends per share (Ad):

Ap = m * Ad

Accelerations of S&P 500 Average Monthly Index Value with Trailing Year Dividends per Share, SF=9, TS=3, Spanning January 2001 Into Mid-2010 with Futures Data But it's not quite that easy. While we can use this method to determine how much stock prices will change, what we can't do is tell when stock prices will undergo this change. We observed that the timing of when the growth rate of stock prices changed in anticipation of changes in the actual growth rate of dividends per share was not consistent. We saw that stock prices seemed to anticipate subsequent actual changes in dividends by anywhere from 1 to 7 months. (The chart to the left shows the change in the dividend growth rate shifted 3 months earlier in time.)

What's more, when we looked at other periods of stock market history, the apparently constant amplification factor was itself not constant. For example, in looking at the period of the 1980s and early 1990s, we found a value of 18 better correlated dividend changes with stock prices, in contrast to the lower value of approximately 9 that we found for the period since 2001.

So, it would appear that where the stock market is concerned, we have a system that operates in a pretty chaotic fashion, with stock prices following the signal sent by changes in the expected future growth rate of dividends per share. Fortunately, we have a name for that: Chaos! In the words of Wikipedia:

Chaotic systems are systems that look random but aren't. They are actually deterministic systems (predictable if you have enough information) governed by physical laws, that are very difficult to predict accurately (a commonly used example is weather forecasting).

More specifically, we call the kind of behavior that we've now observed in the stock market deterministic chaos.

But it's not quite that simple either. The stock market is the venue through which millions and millions of participants interact with one another as billions upon billions of transactions are conducted. Consequently, the stock market can be a pretty noisy place and in fact, noise is always present.

If we consider the change in the expected growth rate of dividends in the future to be the signal driving changes in the price of stocks, noise would consist of any transaction resulting from an investment decision not tied to that signal. For example, various investors might follow a calendar-based investment strategy, or they conduct trades to minimize their tax exposure, to settle the requirements of property division in a divorce, etc.

From time to time, the level of noise in the market can even overwhelm the dividend signal as market participants disregard it altogether: the Dot-Com Bubble of April 1997 through June 2003 perhaps being a classic example.

So how much of the change in stock prices is driven by signal and how much by noise?

Writing in A Random-Walk or Color-Chaos on the Stock Market? Time Frequency Analysis of S&P Indexes in 1996, Ping Chen found in looking at the S&P 500 monthly index (FSPCOM) from 1947 to 1992 that (emphasis ours):

The reconstructed HPCg time series reveals the degree of deterministic approximation of business fluctuations: The correlation coefficient between the filtered and original series is 0.85.

Their ratio of standard deviation, h, is 85.8% for FSPCOM. In other words, about 73.7% of variance can be explained by a deterministic cycle with a well-defined characteristic frequency, even though its amplitude is irregular. This is a typical feature of chaotic oscillation in continuous-time nonlinear dynamical models.

Elsewhere in the paper, Chen observes the approximate average period of oscillation to be approximately 3.6 years, coinciding with the average duration of business cycles.

Chen's work is really remarkable when you consider that he performed his time frequency analysis without knowing what the signal driving those changes in stock prices really looks like, using the phrase "color chaos" to contrast the changes in stock prices driven by apparently deterministic forces with a harmonic period against those driven by "random-walk," or rather, "white noise" factors. What's really cool is that we found what those determistic forces would appear to be and how they would appear to operate.

We'll conclude by sharing Chen's phase space portraits of the historic monthly S&P 500 index value, which reveal his discovery of both the chaotic strange attractors (underlying order with a cyclic pattern) and the type of noise apparent in the S&P 500's stock prices:

Chen, 5a - unfiltered phase portrait S&P 500 Monthly Index Values, 1947-1992 Chen, 5b - filtered phase portrait S&P 500 Monthly Index Values, 1947-1992 Chen, 5c - high frequency noise in S&P 500 Monthly Index Values, 1947-1992
(5a). FSPCOMln HPc unfiltered series. T=60. Some pattern is emerged behind a noisy background. (5b). FSPCOMln HPc filtered series. T=60. Clear pattern of strange attractor can be observed. (5c). FSPCOMln FD series. T=40. The cloud-like pattern indicates the dominance of high frequency noise.

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