I did a similar exercise for the market’s biggest moves from late 2001 through March 2007 and found similar results (see exhibit 34.2). The press sounds a lot like a split-brain patient making up a cause for an effect, and we investors lap it up because the link satisfies a very basic need.
EXHIBIT 34.1 Top 30 S&P 500 Index Moves, 1941-1987
DatePercent ChangeExplanation
10/19/1987 —20.47 Worry over dollar decline and trade deficit; fear of U.S. not supporting dollar
10/21/1987 9.10 Interest rates continue to fall; deficit talks in Washington; bargain hunting
10/26/1987 —8.28 Fear of budget deficits; margin calls; reaction to falling foreign stocks
9/3/1946 —6.73 “No basic reason for the assault on prices”
5/28/1962 —6.68 Kennedy forces rollback of steel price hike
9/26/1955 —6.62 Eisenhower suffers heart attack
6/26/1950 —5.38 Outbreak of Korean War
10/20/1987 5.33 Investors looking for “quality stocks”
9/9/1946 —5.24 Labor unrest in maritime and trucking industries
10/16/1987 —5.16 Fear of trade deficit; fear of higher interest rates; tension with Iran
5/27/1970 5.02 Rumors of change in economic policy. “The stock surge happened for no fundamental reason”
9/11/1986 —4.81 Foreign governments refuse to lower interest rates; crackdown on triple witching announced
8/17/1982 4.76 Interest rates decline
5/29/1962 4.65 Optimistic brokerage letters; institutional and corporate buying; suggestions of tax cut
11/3/1948 —4.61 Truman defeats Dewey
10/9/1974 4.60 Ford to reduce inflation and interest rates
2/25/1946 —4.57 Weakness in economic indicators over past week
10/23/1957 4.49 Eisenhower urges confidence in economy
10/29/1987 4.46 Deficit-reduction talks begin; durable goods orders increase; rallies overseas
11/5/1948 —4.40 Further reaction to Truman victory over Dewey
11/6/1946 —4.31 Profit taking; Republican victories in elections presage deflation
10/7/1974 4.19 Hopes that President Ford would announce strong anti-inflationary measures
11/30/1987 —4.18 Fear of dollar fall
7/12/1974 4.08 Reduction in new loan demands; lower inflation previous month
10/15/1946 4.01 Mean prices decontrolled; prospects of other decontrols
10/25/1982 —4.00 Disappointment over Federal Reserve’s failure to cut discount rates
11/26/1963 3.98 Confidence in Johnson after Kennedy assassination
11/1/1978 3.97 Steps by Carter to strengthen dollar
10/22/1987 —3.92 Iranian attack on Kuwaiti oil terminal; fall in markets overseas; analysts predict lower prices
10/29/1974 3.91 Decline in short-term interest rates; ease in future monetary policy; lower oil prices
Source: Cutler, Poterba, and Summers, “What Moves Stock Prices?” 8. Reproduced with permission.
Investor Risks
As this discussion illustrates, investors should be wary of explanations for market activity. Investors that actively seek explanations for the market’s moves risk one of two pitfalls.
The first pitfall is confusing correlation for causality. Certain events may be correlated to the market’s moves but may not be at all causal. In one extreme example, Cal Tech’s David Leinweber found that the single best predictor of the S&P 500 Index’s performance was butter production in Bangladesh.7 While no thoughtful investor would use butter production for predicting or explaining the market, factors that are economically closer to home may also suggest faulty causation.
The second pitfall is anchoring. Substantial evidence suggests that people anchor on the first number or piece of evidence they hear to explain or describe an event. In one example, researchers asked participants to estimate the percentage of African countries in the United Nations. But before answering, the participants watched the research leader spin a wheel of fortune numbered one to one hundred. When the wheel landed on ten, one group of participants guessed 25 percent. When the wheel landed on sixty-five, another group guessed 45 percent.8 This example may appear frivolous, but investors make serious financial decisions under the influence of similar anchors.
EXHIBIT 34.2 Top 30 S&P 500 Index Moves, September 2001-March 2007
DatePercent ChangeExplanation
07/24/2002 5.73 Investment community decides market overdue for at least a short-term rally; Congressional agreement on corporate-reform law
07/29/2002 5.41 Sense among investors that stocks have fallen too far
09/17/2001 —4.92 First day of trading following 9/11
10/15/2002 4.73 Better-than-expected corporate profits send stocks surging for fourth straight day
09/03/2002 —4.15 Market declines in Europe and Japan and weak U.S. and European manufacturing numbers; talk of more problems among Japanese banks
08/14/2002 4.00 Money moves from bonds to stocks; relief certification deadline passes, and short covering
10/01/2002 4.00 Positive earnings news; Iraq’s agreement to let U.N. inspectors return, and strong economic news
10/11/2002 3.91 Another surge in Chicago Board Options Exchange volatility and short covering
09/24/2001 3.90 Foreign markets (except Japan) report gains; clear optimism in insurance and energy sectors; reduced fear of terrorism; and short covering
07/19/2002 —3.83 Continuing concern about accounting profits
05/08/2002 3.75 A gentle hint from Cisco Systems about a possible coming business recovery is enough to spark a monster stock rally
07/05/2002 3.67 Short covering
03/17/2003 3.54 News that the White House has dropped its sputtering diplomatic efforts and appears to be preparing for war with Iraq
03/24/2003 —3.52 Fears that the war in Iraq could be longer and more difficult than investors had anticipated
10/10/2002 3.50 Short covering; The Chicago Board Options Exchange’s volatility index pushes above fifty—reflects exaggerated level of investor worry
02/27/2007 —3.47 Concern over high Chinese stock valuations and decision by People’s Bank of China to drain liquidity from banking system cause strong sell off in Chinese market; spills over globally
03/13/2003 3.45 United States expresses a willingness to delay until the following week a vote of using force to disarm Iraq
08/05/2002 —3.43 Weaker-than-expected U.S. employment report
07/10/2002 —3.40 Waning confidence in the market and in corporate integrity
01/02/2003 3.32 Anticipation of increased corporate spending; announcement that Bush’s economic stimulus package will be released the following week
07/22/2002 —3.29 Bush affirms support for Treasury Secretary Paul O’Neill and takes some potshots at Wall Street
08/08/2002 3.27 Fed schedules monetary-policy meeting; IMF $30 billion bailout of Brazil; and Citigroup announces a series of corporate-governance measures
09/27/2002 —3.23 Lack of consumer confidence and negative earnings news
09/20/2001 —3.11 Political and economic uncertainty
09/19/2002 —3.01 Bad corporate news and housing construction falls for third straight month
08/06/2002 2.99 Anticipation of interest-rate cut
08/01/2002 —2.96 Report shows slowed manufacturing growth; unemployment worsening; government revises economic growth rates down
01/24/2003 —2.92 North Korea’s nuclear threat; Mideast instability; the war against terrorism and rising tensions with European allies
06/17/2002 2.87 Bargain hunting in tech sector due to an oversold market
01/29/2002 —2.86 Accounting questions surface at more big companies
Source: Wall Street Journal, New York Times, author analysis.
The stock market is not a good place to satiate the inborn human desire to understand cause and effect. Investors should take nonobvious explanations for market movements with a grain of salt. Read
the morning paper explaining yesterday’s action for entertainment, not education.
35
More Power to You
Power Laws and What They Mean for Investors
In the last few years the concept of self-organizing systems—of complex systems in which randomness and chaos seem spontaneously to evolve into unexpected order—has become an increasingly influential idea that links together researchers in many fields, from artificial intelligence to chemistry, from evolution to geology. For whatever reason, however, this movement has so far largely passed economic theory by. It is time to see how the new ideas can usefully be applied to that immensely complex, but indisputably self-organizing system we call the economy.
—Paul Krugman, The Self-Organizing Economy
Zipf It
Here’s an activity to offset ennui on a rainy afternoon. Take a text—say, James Joyce’s Ulysses—and for all the words plot the rank (from the most widely used words to the least-used) and frequency (how often each word occurs).1 If you express this word distribution on a proportional log scale, you will find a straight line from the upper left hand of the chart to the bottom right hand of the chart.2
George K. Zipf, a Harvard linguist, noticed this relationship in a number of systems in the 1930s and summarized them in his famous book Human Behavior and the Principle of Least Effort. Zipf’s law, as scientists came to call it, is actually only one example among many of a “power law.” To take language as an example, a power law implies that you see a few words very frequently and many words relatively rarely.
Zipf erroneously argued that his law distinguished the social sciences from the physical sciences. Since his work, scientists have discovered power laws in many areas, including physical and biological systems. For example, scientists use power laws to explain relationships between the mass and metabolic rates of animals, frequency and magnitude of earthquakes (the Gutenberg-Richter law), and frequency and size of avalanches. Power laws are also very prominent in social systems, including income distribution (Pareto’s law), city size, Internet traffic, company size, and changes in stock price. Many people recognize power laws through the more colloquial “80/20 rule.”3
Why should investors care about power laws? First, the existence of power law distributions can help reorient our understanding of risk. Most of finance theory—including models of risk—is based on the idea of normal or lognormal distributions of stock price changes. A power law distribution suggests periodic, albeit infrequent price movements that are much larger than the theory predicts. This fat-tail phenomenon is important for portfolio construction and leverage.
Second, the existence of power laws suggests some underlying order in self-organizing systems. Even though scientists haven’t fully explained the mechanisms that lead to power laws in social systems, we have enough evidence that power laws exist to make some structural predictions about what certain systems will look like in the future.
Finally, standard economic theory does not easily explain these power laws. For example, neoclassical economics focuses on equilibrium outcomes and assumes that individuals are fully informed, rational, and that they interact with one another indirectly (through markets). In the real world, people are adaptive, are not fully informed, and deal directly with one another. So ideally we should seek to explain the empirical findings with an approach that fits how people really act.4
The More Things Change . . .
Zipf specified a very simple equation to express his law:Rank x Size = Constant
This equation says that the quantity under study is inversely proportional to the rank. Given Zipf’s equation, we can obtain a sequence by multiplying the constant by 1, 1/2, 1/3, 1/4, etc. Take the case of city-size distributions in Spain. If the largest city, Madrid, has 3 million inhabitants, the second-largest city, Barcelona, has one-half as many, the third-largest city, Valencia, one-third as many, and so forth. Zipf’s law does describe some systems well, but is too narrow to describe the variety of systems that exhibit power laws.
The brilliant polymath Benoit Mandelbrot showed that two modifications to Zipf’s law make it possible to obtain a more general power law.5 The first modification is to add a constant to the rank. This changes the sequence to 1/(1 + constant), 1/(2 + constant), 1/(3 + constant), etc.
The second modification is to add a constant to the power of 1 in the denominator. This yields 1/(1 + constant)1 + constant, 1/(2 + constant)1 + constant, etc. The modified power can be a whole number or an intermediate value (e.g., 1/(1 + constant)3/4). Zipf’s law is the special case where both constants are set to zero.
Even with the introduction of these two parameters, the generalization from Zipf’s law to a broader set of power laws remains very simple. That such an elementary equation describes such diverse phenomena certainly evokes wonder, especially since we have no unified explanation for how these power laws come about.
One of the interesting features of power laws in social systems is their robustness. For example, exhibit 35.1 shows the plot for the rank and size in U.S. cities from 1790 to 1990. Notwithstanding population growth and substantial geographic shifts, the relationship between rank and size remained remarkably consistent for 200 years.
Another example, and more directly applicable for investors, is company size. Exhibit 35.2 shows that the relationship between sales and frequency for U.S. companies in 1997 follows Zipf’s law. Economist Rob Axtell created this chart based on U.S. Census Bureau data, which were not available until early 2001, based on 5.5 million firms and more than 100 million employees.
Axtell notes that the distribution of firm sizes is insensitive to changes in political and regulatory environments, waves of mergers and acquisitions, new firm and bankruptcy trends, and even large-scale demographic transitions within the workforce (e.g., women entering the U.S. workforce).6 The implication is that there are important underlying mechanisms that create the order we see.
EXHIBIT 35.1 Rank and Size of U.S. Cities, 1790-1990
Source: Batten, Discovering Artificial Economics, 165. Reproduced by permission of Westview Press, a member of Perseus Books, L.L.C.
No one completely understands the mechanisms that yield power laws, but there are a number of models or processes that generate them.7 Perhaps the best known is “self-organized criticality”—a model popularized by theoretical physicist Per Bak. Bak suggests a scene where a child is at a beach letting sand trickle down into a pile. At first the pile is relatively flat and the grains remain close to where they fall. Once the pile becomes steeper, additional grains will periodically trigger a little sand slide. A while longer and the sand slides will be as big as the pile itself. The system is in a “critical” state—between steady state and randomness. Once the pile is in a critical state, additional grains produce sand slides of varying magnitudes and the sizes of the sand slides follow a power law.8
EXHIBIT 35.2 Sales and Cumulative Probability for U.S. Firms, 1997
Source: Axtell, “Zipf Distributions of U.S. Firm Sizes,” 1819. Reproduced with permission.
There are aspects of the sand-pile metaphor that are useful for thinking about social systems. For one, the economic systems are clearly self-organizing. That is, most companies, cities, and countries are the result of interactions among individuals, not of central planning. Also, there is a sense of a critical state. In a physical system, a critical point is one where a small change produces a phase transition—for example, water freezes as the temperature drops below zero degrees centigrade. Economists do not define critical points as clearly for economic systems, but we do know that individuals neither stay at the same company forever (steady state) nor haphazardly jump from company to company (randomness). Axtell has captured these features through an agent-based model to explain firm and city sizes. His model yields results that are consistent with the empirical data.9
Catch the Power
There are a number of ways that an understanding of power laws helps investors. The first way builds off
Axtell’s work on company size. Given the evidence that power law distributions are robust over time, we have a good sense of what the distribution will look like in the future even though we have no idea where individual companies will fall within it.10 But given reasonable assumptions for economic growth and inflation, we can derive a good estimate of the probabilities of companies being of a particular size.
We know ahead of time, for example, that a miniscule percentage of companies will be very large (e.g., > $200 billion sales). We can look at the imputed growth rates of large companies today and discern how many of them are projected, based on expected growth, to be very large. If the group projected to be very large vastly exceeds the percentage that will be large, we know there is the likelihood of substantial downward expectation revision.
Another way investors can use power laws is to understand the topology of the Internet. A classic example of a self-organizing network, the Internet has spawned a host of power law relationships—including the number of links per site, the number of pages per site, and the popularity of sites. These power laws suggest uneven benefits for companies that make heavy use of the Web.11 The development of the Web may be instructive for the organization of future networks.
Power laws represent a number of social, biological, and physical systems with fascinating accuracy. Further, many of the areas where power laws exist intersect directly with the interests of investors. An appreciation of power laws may provide astute investors with a useful differential insight into the investment process.
More Than You Know Page 21
