Theodore Modis

Presented at the 5th International Conference of the Decision Sciences Institute

4-7 July 1999, Athens, Greece.

This article is based on work reported in the author's recent book An S-Shaped Trail to Wall Street, Growth Dynamics, Geneva, 1999.



Under the assumption that competition (Darwinian in nature) reigns in the stock market, we can analyze the growth of company stocks as if they were species competing for investors' resources. The approach requires the study of the dollar value and the share volume daily exchanged at the stock market. These two variables-contrary to prices-obey the law of natural growth in competition, which like every natural law, is endowed with predictability. The forecasts indicate that whereas there is no looming crash, no significant growth should be expected either. The DJIA is to hover between 8500 and 9500 for a few years.



Every now and then proponents of the exact sciences bring their trade to the stock market. In the 1980s we saw the so-called rocket scientists (e.g., the Prediction Company), and chaos scientists (e.g., B. Mandelbrot) try their hand, but no one broke the bank. As the shine of chaos and fractals wore off, complexity came into the limelight, bringing along admixtures from biology and ecology. The new approach looks at a bigger picture by paying attention to the law of competition. But the object of most studies at the stock market continues to be the squiggly charts of stock prices, and market indices. These indicators do not constitute competition variables because they do not represent limited resources. The fact that the IBM stock goes up does not prevent the AT&T stock from also going up. Competition emerges only when we deal with a limited resource and one competitor's acquisition entails another competitor's nonownership.



There are competition variables at the stock market. Some of them even occupy prime place in the daily news reports, but receive less attention than stock prices and market indices.

Value and Volume

The amount of dollars (value) and the number of shares (volume) exchanging hands daily, constitute limited resources. The money spent in buying one stock is no longer available for acquiring another stock. Similarly, once an investor buys a share, this share is no longer available to other investors.

The daily volume reflects the attention investors paid to that stock that day. Active stocks sometimes attract considerable attention even from non-investors. They receive special mention and may be highlighted in more than one article in the daily newspaper. The value exchanged over a company's stock in a day is less visible but unquestionably reflects the extent to which the stock tied up investors' dollars that day. If money and attention go to one stock, they do not go to other stocks. Unlike price, these variables do represent limited resources.

Volume and value obey the law of competition directly. Prices come into the picture only indirectly, and so do the various price-based indices, such as, the Dow Jones Industrial Average (DJIA). Many analysts and forecasters have long slaved over curves and historical data of the DJIA searching for structures, periodicities, waves, or other observations that would help anticipate future patterns. Patterns in the evolution of the DJIA sometimes correlate with patterns in the evolution of the share volume and the dollar value. But if we want to quantitatively study growth in competition at the stock market, via such mathematical formulations as the logistic equation, and the Volterra-Lotka system of equations, we need to concentrate on the variables that intimately relate to competition and its fundamental mechanism.

Market Share

Another competitive variable is market share, this archetypal measure of a company's competitive performance. If the market share is increasing, the company is most likely enjoying competitive advantages. Market share constitutes par excellence a limited resource. One competitor's gains in market share are necessarily at the expense of one or more of the other competitors. But often it is not specified to what market share we are referring. When we read that Apple's market share is 20 percent of the personal computer (PC) market, it is not always clear whether 20 percent of all PC buyers buy Apple computers, or whether 20 percent of all expenditures on PCs go to Apple. The difference can be important.



If all products in a market had the same price, market shares in units would be identical to market shares in dollars. But because prices vary, we find ourselves in a world in which there are two kinds of competition: Competition for consumers' dollars, and competition for consumers' needs. The performance of the competing companies can be rated via their market share in dollars and their market share in units. The two market shares can be quite different. Suppose, for example, that the luxury-car market in America consists of only two car manufacturers, Rolls Royce and Lincoln Continental, the former priced at $200,000 and the latter at $60,000. The market share in units of Rolls may be 25 percent while its market share in dollars, due to higher prices, could be 40 percent. Such a differentiation originates with market shares, but shapes activities in the departments of marketing, advertising, price-setting strategies, and other departments of the competitors' organizations.

Market shares represent probabilities. They dictate what will happen on the average over large numbers of transactions. It would be mathematically correct to say that when the average oil-rich sheik goes shopping for a luxury car in America, he will write a check 40 percent of which will be for a quarter of a Rolls Royce, and the remaining 60 percent for three quarters of a Lincoln Continental. But this may be the situation today. If our sheik comes back next year, the market shares may have changed significantly as a consequence of the fact that competitive advantages vary with time. The evolution of market shares describes the evolution of the competitive struggle. Any difference between the market share in units and the market share in dollars is due to differences in price among the competitors.

At the Stock Market

There are two kinds of competition at the stock market too, competition for dollars (value) and competition for attention (volume). Here again the variables for each stock can be expressed as percentages. One variable would be a percentage of the total daily NYSE share volume, and the other a percentage of the total daily NYSE dollar value. Numerically these percentages are equal to the probability that the average investor buys the stock, and the probability that his or her money becomes invested in the stock, respectively. The market share in units indicates the probability that one share is sold, and the market share in dollars indicates the probability that one dollar is invested in this stock.

With other things being equal, when you call your stockbroker with the intention to invest some money, the probability that you will end up buying a particular stock is not equal to the probability that one of your dollars will be invested in that stock. We are only speaking statistically, of course, which is what is behind the phrase, other things being equal. You may argue that when you have made up your mind about an investment, other things are not at all equal, and that all your dollars will end up on the stock you have chosen. This is true, just as it is true for every other investor who makes similar moves. But that is how statistics works. When the 50-odd million American investors place their dollars in the 4,000-odd stocks on the Big Board, you can be sure that the laws of statistics, and the laws of natural competition will be honored. These are the laws that can be exploited in order to see more clearly into the future.

The two kinds of competition serve like two dimensions which taken together constitute a powerful instrument for understanding the behavior of stocks in the future. Like stereoscopic vision that permits one to focus at a certain depth in space, one also needs two different points of view to see clearly at a certain depth in time.



Populations of cells follow S-shaped patterns as they grow, just as do populations of rabbits. But cell populations can constitute a multicellular organism, which will also grow along S-curves. A group of people linked together with a common goal, interest, ability or affiliation, for example, selling products in order to make a profit, in other words, a company or an organization, can behave as one individual. The individuals now play the role of the cells in a multicellular organism, and their assembly becomes the organism. The growth of an organization, therefore, may be expected to follow the familiar S-shaped pattern we have seen for individuals.[1] Raising the abstraction level, we may want to view a group of companies that share much common fate, such as the 30 industrials of the Dow Jones, to behave as single company thus also adhering to S-shaped patterns.

To forecast the Dow Jones Industrial Average (DJIA) one must fit a logistic curve to the historical evolution of the total volume, and independently another one to the total value, daily exchanged over the 30 industrials. The volume here must be the true-share volume; that is, the volume corrected for stock splits. This way the ratio of the two logistic curves will directly yield the DJIA.

This was done and the results are shown in Figure 1. Ten years' worth of daily quotes for the 30 industrial companies of the DJIA were first corrected for stock splits, and then used to calculate two time series (dollar value and share volume) for the Dow as a single entity. The exercise was repeated with the two time series constructed as fractions of the NYSE totals. It gave consistent results, not shown in the figure.

The reader must remember that the smooth lines in the figure do not represent fits on the data points shown. Each smooth line represents the ratio of two logistics fitted on the value and volume data sets of the respective historical widows. The agreement between data and model description is better than the figure seems to indicate. The model description refers to a DJIA not quite the same as the official one charted in the figure. The procedure of diving value by volume yields a price average weighted by the volume, whereas the official DJIA follows form a simple arithmetic average of the prices. Despite general agreement between these two averages, the weighted average tends to fall below the arithmetic one in recent years, because several relatively high priced stocks sell in relatively small numbers (e.g., J. P. Morgan, Merck, Chevron, and 3M).

The Competition Model for the Dow Jones

Figure 1. A study made on October 16, 1998. The data (irregular line) are daily quotes for the DJIA. The thick smooth line is a description based on the last ten years. The thin smooth line is a description based on the preceding 18 months. (Remark: The DJIA forecasts are calculated from the ratio dollar value / share volume. This ratio can vary significantly despite the relative stability of the two terms, therefore there is an imperative need for frequent updates as discussed in An S-Shaped Trail to Wall Street.)


The description based on the last ten years of history produces a long-term forecast. It indicates low growth in the near future with a possible decline during the early 21st century. It represents a coarse general trend and misses sizeable excursions, up or down, such as, the market correction in September 1998 and the recovery in early 1999.

The description based on the historical window April 1997 to October 1998 provides us with a shorter-term forecast that follows the market more closely through February 1999, the time of this writing. It indicates that the DJIA should cross the 9000 level definitively only in early 1999.

It follows from both forecasts that whereas there is no ominous crash of the stock market in sight, there is no significant growth either. The DJIA is expected to hover between 8500 and 9500 for a few years. But as with every period of stagnation, large fluctuations of up to 15% would be compatible with the chaotic character of such a "winter" season, which may well have began in mid 1997.[2] Such a "pessimistic" forecast for the NYSE receives credibility from the fact that spending already exceeds earning in the American economy.



The assumption that company stocks compete for investors' resources in a Darwinian way led to the study of the variables dollar value daily exchanged, and share volume daily exchanged, instead of closing stock prices. Fitting logistic growth functions to the evolution of these two variables yielded forecasts for the DJIA. The added value of this approach, compared to approaches based directly on prices, stems from the fact that the variables we are studying-value and volume-obey the law of natural growth in competition. And that law, like every natural law, is endowed with predictability.

Both the long-term and the short-term forecasts indicate that the DJIA is not expected to grow in the coming years. It should hover, possibly with large fluctuations, between 8500 and 9500.


[1] Modis, T. (1992), Predictions, Simon & Schuster, New York.

[2] Modis, T. (1998), Conquering Uncertainty, McGraw-Hill, New York.