January 28, 2009

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):

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:

(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.

Labels: ,

Unexpectedly Intriguing!

is good for you

Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at:

ironman at politicalcalculations.com

Recent Posts

Highly Questionable Higher Education Costs

China in Recession

On the Moneyed Midways - January 23, 2009

Markets in Everything: Obama Action Figure

Waiting for Howard Silverblatt

The Economic Detective: A Final Gruesome Discovery...

On the Moneyed Midways - January 16, 2009

The Economic Detective: Prime Suspect Revealed

The Economic Detective: Lining Up the Suspects

The Economic Detective: Victim Autopsy

Elsewhere on the Web

This year, we'll be experimenting with a number of apps to bring more of a current events focus to Political Calculations - we're test driving the app(s) below!

Most Popular Posts
Quick Index

U.S. GDP Temperature Gauge

Political Calculations' U.S. GDP Temperature Gauge provides a means to quickly evaluate the growth rate of the U.S. economy against the backdrop of how the economy has performed since 1980, with the "temperature" color spectrum ranging from a recessionary "cold" (purple) through an expansionary "hot" (red).

The GDP Temperature Gauge presents both the annualized GDP growth rate as reported by the U.S. Bureau of Economic Analysis reports for a one-quarter period and also as averaged over a two quarter period, which smooths out the volatility seen in the one-quarter data and provides a better indication of the relative strength of the U.S. economy over time.

Site Data

Visitors since December 6, 2004:

#### JavaScript

The tools on this site are built using JavaScript. If you would like to learn more, one of the best free resources on the web is available at W3Schools.com.

#### Other Cool Resources

ZunZun - Exceptional regression analysis tool.
Wolfram Integrator - Solve integrals. Do calculus!
Create a Graph - Easy-to-use basic graph-making tool.
Many Eyes - Data visualization extraordinaire!
Wolfram Alpha - Computational knowledge engine.
Khan Academy - Math & science video mini-lectures!
Picasion - Animate images.

Archives
December 2004
January 2005
February 2005
March 2005
April 2005
May 2005
June 2005
July 2005
August 2005
September 2005
October 2005
November 2005
December 2005
January 2006
February 2006
March 2006
April 2006
May 2006
June 2006
July 2006
August 2006
September 2006
October 2006
November 2006
December 2006
January 2007
February 2007
March 2007
April 2007
May 2007
June 2007
July 2007
August 2007
September 2007
October 2007
November 2007
December 2007
January 2008
February 2008
March 2008
April 2008
May 2008
June 2008
July 2008
August 2008
September 2008
October 2008
November 2008
December 2008
January 2009
February 2009
March 2009
April 2009
May 2009
June 2009
July 2009
August 2009
September 2009
October 2009
November 2009
December 2009
January 2010
February 2010
March 2010
April 2010
May 2010
June 2010
July 2010
August 2010
September 2010
October 2010
November 2010
December 2010
January 2011
February 2011
March 2011
April 2011
May 2011
June 2011
July 2011
August 2011
September 2011
October 2011
November 2011
December 2011
January 2012
February 2012
March 2012
April 2012
May 2012
June 2012
July 2012
August 2012
September 2012
October 2012
November 2012
December 2012
January 2013
February 2013
March 2013
April 2013
May 2013