The S-curve, which is mainly used in population studies, is now redefining business strategy and stock market forecasting.

“The history of the world is nothing but the biography of great men,” said Thomas Carlyle, the 19th century historian. It’s strange but the more we try to understand markets, the more it pushes us back to econohistory. In March this year we had written about Thomas Malthus (1766-1834), economic history’s greatest pessimist who talked about hunger and starvation. Though the science of forecasting is still young and underdeveloped, Malthus made an amazing forecast of a crisis by the middle of the 19th century. And his population studies are turning things upside down more than 150 years after his death.

In 1838 after reading Malthus’s essay on the principle of population Pierre F Verhulst, a Belgian mathematician, published the Verhulst equation. However, it was only in the 20th century that Alfred Lotka of Johns Hopkins University and Vito Volterra of University of Rome generalised Verhulst’s growth equation to model competition among different species.

These are the origins of the S-curve. S-curve fitting, a natural and fundamental approach to forecasting, is reliable with high confidence levels. Physicists have shown that everything in nature can be quantified, from matter to light. Spectacular consequences of putting natural law descriptions in a discrete form have been the subject of chaos and fractals. I had a chance to meet Theodore Modis, a physicist from Growth Dynamics Inc, recently in Vienna. Along with Alain Debecker, a mathematician from Lyon University, he wrote about the S-curve and the bridge between continuous and discrete formulations spanning 150 years of developments in mathematics. They also wrote about how it starts with Verhulst and finishes with Mandelbrot, intricately linking order with chaos. The paper also mention how chaos-like states can be expected before and after logistic growth ie historical picture is nothing but an alternation of logistic growth with periods of instability. The chaotic fluctuations belong to the end of a growth phase as much as to the beginning of the next one. A well established S-curve will point to the time when chaotic oscillations should be expected. What’s more interesting about this paper is that it sites the Kondratieff cycle (56 years) as a way to position growth periods.

According to the S-curve, society pushes natural growth factors to an invariant status such as income spent on travelling at 15 per cent, sleep to work ratio of eight hours, mammal heart beats of 1 billion in a life, hospital infections at 14 per cent, average car speed at 30 miles per hour etc. These invariants happen as respective growth curves hit respective ceilings. In competing products, these ceilings and invariant status can also explain substitution. For example, when wood usage hits a ceiling, coal takes over; as coal fails, oil takes over and as oil will exhaust, it will be substituted by natural gas and so on. There are some rules to the S-curve growth. It proceeds in stages and each stage represents the filling of a niche with limited capacity. And just like economic growth, political growth also shows alternation between order and chaos. According to the curve, logistic growth is natural growth in competition.

Modis has extended the S-curve model to stock markets assuming stocks to be species competing for investor resources. He trashes the Gaussian bell curve since there is no natural law behind it and suggests that all marketers using bell shaped curve for strategy are headed for failure. Cyclicality can give strategy answers regarding cannibalisation and future growth. The author also junks the goodness of exponential fits and proves how correlation does not imply good fit. Exponentiality according to him means extrapolation on a log scale, which can’t predict. He also goes ahead and says that a pattern can be used to make forecasts, as long as it represents a natural law that guarantees invariability.

According to Modis, volume and value obey the law of competition directly. He also made some bold prediction on the Dow Jones starting June 2008. He predicts prices not higher than 14000 with lower targets lying at 8000. Other forecasts are about world population, which he claims should have a final ceiling at nine billion people, cumulative oil production in the US should hit a ceiling at 220,000 million barrels by 2030, Microsoft needs to undergo a major change for survival and the next energy growth assets should be natural gas and nuclear power. The substitution aspect of the curve is clearly cyclical and it seems we have no choice but to move to renewable energy source after hydrogen nears a ceiling on the S-curve.

Despite a thorough track record and mathematical grounding, the S-curve suffers from a few kinks. It really does not account for any other fractal apart from the S-curve. It does not take into account Fibonacci numbers or ratios, seems more for investment than trading, has a clear disbelief on price patterns, looks for parameters that intimately relate to competition and fundamental mechanism, saw the post-1999 period as one for stagnation than the one for crash, accepts sunspot cyclicality as a good predictor but not fractals or Elliott Wave, which are cyclical by nature.

Walter E White’s contribution in 1968-70 only reinforces the gaps in the S-curve. White said that Elliott Wave analysis suggests a general relationship between static forms in plants and animal life and dynamic waves of time. The origins of this relationship may be found in fundamental ideas of arithmetic, logic, algebra, geometry and trigonometry dating back to 500 BC. Elliott Wave has a cross-subject application and the idea of shock or chaos is fundamental. White’s contribution quotes Kierkegaard (teacher of Neil Bohr of quantum mechanics fame) saying that “in life only sudden decisions, leaps, or jerks can lead to progress”. All this brings Elliott waves in sync with the chaos and order that mathematicians have been talking about for over 150 years. Above this the relationship between the logarithmic spiral, the Fibonacci series and the golden ratio has been known for about 2,500 years, making Elliott pattern based on a natural law.

What’s strange is that while mathematicians were working on a growth decay natural model, Ralph Elliott was refining Charles Dow’s market fractal theory. The noise against Elliott keeps rising everyday while pure Elliotticians keep defending it. In August 2007, Robert Prechter highlighted the comparisons and improved predictability of the wave principle over Sornette’s log periodic cycles. According to Prechter, Elliott is a science with clear rules though practicing it is a craft.

The S-curve also offers competition to the wave principle. However, with the open gaps and the new school of thought that economics and finance are two different subjects altogether, the challenge is alive. After all, double decimal accuracy forecast delivered by Elliotticians over the last 60 years on all trading time frames and with practitioners like Hamilton Bolton delivering as much as 11 accurate yearly forecasts in 13 years, the S-curve has a tall benchmark to compete. The only research aspect which really suffers with the S-curve gaining ground is the investigative equity research, the last vestige of equity research which still holds some water, while the Fibonacci reality of markets remains a notch ahead of the S-curves.