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Chaos in Business

Enron's collapse

Enron is a classic example of the forces of chaos operating upon a complex system.

Introducing the principle of the share price capacity.

Countrywide Financial is another beautiful example of chaos in action on the financial markets. The warning signs for this started years ago. 




Coca Cola

Walt Disney


 General Electric


 Johnson & Johnson




 Altria Group





Countrywide financial

JPM Morgan & Bear Stearns


Chaos , as a science and a discipline, emerged the 1970's. Theorists employ mathematical tools to describe the dynamic processes found in complex systems, such as ecosystems and the weather. Differential equations of mathematics describe the way systems change smoothly and continuously over time and so we may use them to study dynamic systems. A model is built as "a mathematical construct which, with the addition of certain verbal interpretations, describes the observed phenomena" (Neuman, 1988, in Gleick). How we perceive something is largely dictated by the interpretation.

Amazon.com Widgets

By setting basic parameters of a complex system model at the start of a time sequence we can study the consequence of variations. It is important to be aware of inherent limitations to such systems. These variations are real and need to be taken into account. A businesses share price, for example, is linked to real business factors, such as debt and profits. These in turn influence the investor. If by some accounting process, a false profitability is reflected, investors can push up a share price. However, the company that becomes dependent upon a reflection of  its share price for real growth, will eventually face a dire constraint, as the true debt and profitability are discovered and responded to by investors. A sound company needs to identify and define true economic constraints so that challenges can be met by rational, managed adjustments natural economic factors. This article will use the example of Enron and some principles from the science of chaos to identify some of these.

In looking at complex systems through the lens of chaos science, the time scale is important. A holistic view is needed. Look at too narrow a range of data and patterns reflecting a different set of influences are observable. Someimes, looking too broadly can also hide observable patterns. As such there are patterns within patterns. To discern the health of a company, it is better to look at the history of the whole company. In this article, I look at the share price as a simple reflector of company performance. View the following graph from BBC.com.

Enron share price through 2001 

Image 1: A chaos theorist looking at this would comment that the data may indicate a company "beyond the edge of chaos", but that a wider span of time was needed. None of the raw data, the daily ups and downs of the company, the news reports, business reviews, expert opinions and the like matters too much here. However, there is insufficient information for a full analysis. A soon as the whole history of the company's share price is reflected graphically (Image 2), a chaos theorist is bound to become very excited. Here are symptoms of chaos and such symptoms reflect stress of some type. This stress reflects the fact that the company has exceed the bounds of its natural limits. It indicates a need for an adjustment, either through self-regulation or via external forces.

Enron: share price history

Image 2

Here, a chaos theorist, and anybody for that matter, can see that there was some unusual development at the end of 1999. The sharp share price increase is followed by wild fluctuations, termed chaotic variation. The same pattern can be generated with simple mathematical models,  the basis of which I will now proceed to explain.

We can develop simple models to follow a hypothetical complex business through time. Each day's share price depends on the previous day's, so that the whole history of a business share price becomes available through this process of functional iteration. In a feedback loop, each day's output serves as the next day's input. In non-linear systems there is disorder and erratic changes, which reduce the possibility of deterministic analysis as time proceeds, yet the study of chaos led to the discovery and establishment of what is called, DETERMINISTIC CHAOS. Chaos science shows that simple deterministic models could produce what looked like random behaviour. The behaviour actually had a structure or pattern, yet any piece of it is indistinguishable from noise and so cannot be analysed on its own.

The formula: x(next) =  rx(1-x)

represents a mathematical formula supposed to model complex systems." r" represents the rate at which the share price increases..

 graf1b.jpg x represents a relative share price as a number less than one, where one is the factor representing the maximum share price. Share price is expressed in this formula as a ratio between zero and one, where one is the maximum attainable price. The output from the formula becomes the input for the next cycle, so mirroring how a share price changes. Here is the behaviour of a share price according to the formula in Fig. 1. (x starting at 0.03 and r = 2.9). The share price is supposed to rise rapidly, responding to new investor confidence, overshoot its perceived viable limits, fall back below its maximum capacity and oscillate over this limit until it reaches equilibrium.

 graf2b.jpg When r is increased to three, (Fig. 2) the oscillations do not converge on a stable equilibrium, even after 100 cycles (iterations, days, months).


 graf3b.jpg If r is increased to 3.1 (Fig. 3) there is a definite separation of the oscillations into two "periods". This is precisely the same formula, with only the share growth rate (r) increased!



If r = 3.44 we see new behaviour emerging - the two periods begin to split (Fig. 4). graf5b.jpg

By the time r = 3.5 there are clearly 4 periods (fig. 5). Again the only variable in the formula to change is the share price growth rate (r). The share price is oscillating between four levels in a regular, stable pattern. Note that in each of these graphs, the growth rate of the share price is held constant to reflect a pattern. In the real world, this will vary, but in a statistically predictable way.


    To smooth out the data we now start with x = 0.5 and put r = 3.56 (Fig. 6). Again we see an increase in the number of periods. What is happening? We can interpret this by saying that it is possible that where information is limited, investor uncertainty develops, some speculating on a share price increase, others anticipating a price decrease. This acts as a feedback limiting further share price growth, and the share price will be thrown into some very regular cycles or oscillations that vary with its rate of increase and the capacity or "ceiling price" of the shares. It is important to note two factors involved here, the rate of increase of the share price and some limit or perceived capacity to the actual share price. These two terms, the rate of the share price increase and the share price capacity will be used throughout this article as two chaos parameters or terms. We will also use the term “oscillating between periods” where distinct periods are discernable. Another descriptive term is the presence or lack of “damping”. It is the lack of damping at the share price capacity that leads to chaotic behaviour. Damping is some method of slowing the rate of increase in a share price.

 graf7b.jpg Let's explore this further. By r = 3.6, order is starting to collapse (Fig. 7). (In  the real world changes are random and prevent such regular patterns. In statistics (e.g. regression analysis) they lump these variables  together as  " stochastic  disturbance " or "stochastic error term" (Gujarati, 1988). This disturbance term makes allowance for factors that affect the system, but are not included in the model. Mathematical models are useful in showing the underlying patterns.

 graf8b.jpg By r = 3.7 order is largely lost (Fig. 8).

The system is displaying chaotic behaviour because of the interaction of the share growth rate (r) and the share price capacity (limits) (K = 1) in the model! The share price capacity is established through a combination of real economic factors and investor perceptions.


By r = 3.99 (Fig. 9) the system governed by our simple formula has become chaotic - periodicity has given way to chaos, fluctuations that never settle down (Gleick, 1987). An investor looking at such data from a short period in the stock's performance, would never guess that there is a pattern to this erratic fluctuation in the share price. How could he perceive that the same system, with only slight changes in two parameters, could display such a variety of patterns? Further, stochastic variations from other influences are not even considered here!

 graf10b.jpg At r = 4 the mathematical formula fails, the share price collapses to zero and the company goes bankrupt. The mathematical "environment" could not cope with such a high share price growth rate in relation to the share price capacity, limits (k<1) were greatly exceeded and "extinction" occurred (Fig. 10)!


 Let us try to bring the model's expression closer to the reality of what occurs in business and see how chaos can develop as the rate of increase of the share price escalates. In fig 11 there is a slow incremental increase in the share growth rate (r) over time to illustrate all the above behaviour (Figs. 1 to 10) in one picture. From Fig. 11 we can see that there is an initial, rapid increase in the share price. In this region, the market appear unlimited, so there is no apparent limit to the share growth rate, but only the intrinsic rate of increase of the company. This intrinsic rate of increase is determined by factors such as available funding, or how long it takes to train new employees or to setup branches in other parts of the country. The business has real limits to many such intrinsic factors.

Then the curve flattens as the effect of environmental limits, the share price capacity, comes into effect. This could be termed market saturation, but this would be only one factor determining the share price capacity (spc). Feedback from the perceived or actual limits define the share price capacity (spc). A skilled investor or broker can identify companies that are undervalued through an accessment of the business and its global potential. A such, entry into chaos may reflect a false spc. An investor knowing this will invest with the confidence that a new spc will soon be defined through the market forces of supply and demand. In this case, as the growth rate does not respond sufficiently to the limits defined by the spc, the company's share price explodes into chaos near the share price capacity for the share. At one point extinction occurs. When the growth rate is very high, this happens very quickly.

Take another look at the Enron share price in 1999 (Image 2). The rate of increase is massive, followed by an entry into chaos as the share price capacity is exceeded. Prices fluctuate wildly. In this case, the "theoretically possible, but practically impossible" happens - a massive corporation becomes "extinct". A deeper investigation reveals that the whole share price structure is false and fraudulent. This discovery redefines the share price capacity at a much lower level, and the company collapses into bankruptcy. A strong company would have recovered, starting from a much lower but viable share price, but Enron had hidden billions of dollars in debts and operating losses through complex accounting schemes. Once these became known, investors disappeared.
    In the above model, the share price capacity is a value of one. Growth is smooth until the share price approaches this value and then with sustained growth, suddenly the smooth curve bursts into chaos. This is the region that chaos theorists call the edge of chaos . The formula for this curve is:

x(next) = r(inc)x(1-x)

where r(inc) = previous r value + 0.005. With each iteration of the formula the rate of increase (r) increases by 0.005.

 graf12b.jpg If the share price reflects the real situation within the business, as the share price approaches the share price capacity, the share price growth rate (r) should decrease. Clearly, the share price growth rate is subject to environmental constraints. The company has an accounting "book value" that defines something real. Thus, as the share price (x) approaches one (in the model), the share price growth rate (r) must decline. This is termed “damping”. This aspect needs to be built into a management model to prevent a company crossing the edge of chaos and having its share price go chaotic. The company should respond to its environment (feedback) and dampen (regulate) the share price if conditions tend to become chaotic. If the company does not do this, usually a new, lower share price capacity is defined (see Microsoft's share price history). This environmental change takes on many forms, from a loss of investor confidence (a perceived limit) to a loss of market share (a real limit). By inserting ("building ") this dampening condition into our formula, we find that the population growth never explodes into chaos (Fig. 12), as it does not cross the edge of chaos. A company needs to investigate and discover its own limits and define some type of parameters that come into play to dampen (limit) the rate of the share price increase when the edge of chaos is approached. This proactive management will stabilise the stock performance of any company. Here there is a rapid feedback from environmental or market conditions followed by a response from the company through the application of dampening measures that come into play to slow the share price growth rate.

A situation can occur where investor confidence suddenly disappears and a share growth rate (r) share price is suddenly too high. This happened to Enron. In effect, a sudden resource limit (shareholder investment), throws the population into chaos. Companies can recover from this, if the correct dampening measures are applied. We see the impact a loss of shareholder confidence upon the share price rate(r) of increase in Fig. 13. Chaos smoothes out to a new stable situation, the epitome of  holism. Here, the initial growth rate was four, placing the population at the edge or in total chaos. I calculated this rate as:

r(next) = r + 0.005 - x/100. As the share price (x) approaches the spc, there is then a negative impact on the share price growth rate (r).


Data from a business that appears chaotic as in the beginning of Fig. 13 may be reflecting stress due to a growth rate that is too high in relation to the constraints (available resources) of the business environment. Market forces quickly reduce the share price growth rate through slowed investment. If businesses do not respond to financial or resource limits, by regulating their growth rate, they may go bankrupt!  In the example here (Fig. 13), the company is fundamentally sound (honest) and viable, so the adjustment stabilises.

This is now a well recognised principle in chaos studies. Refer for example to http://www.abdn.ac.uk/physics/s6/Chaossum.pdf (2005). Here Dr Neilson introduces the same formula that I illustrated over 10 years ago. One added term that emerges from his examples below is the increasing amplitude of the oscillations as chaotic behaviour is approached.

The inserted graph reflects the behaviour of historic (time based) share price variation that is illustrated in this section on chaos in business. A simple oscillation has an initial growth phase, but then stagnates into a stable value at the share price capacity. Such a company is surviving, but does not present any growth opportunity for an investor.

Period doubling is where the share price oscillates between two distinct values without any effective damping to moderate or impact on this fluctuation.


When the period quadruples the share price clearly oscillates between four highs and lows. As the amplitude of these oscillations increases, another split occurs.

This is a highly unstable share at the very edge of chaos. It is a very undesirable scenario. At this point management should urgently take measures to dampen the rate of increase of the share price.
Here the share price has gone chaotic. The executive management, the directors and the financial advisors have lost control of the company. The company is at risk of total collapse.

Future trends

Managers at banks, brokerage houses, and hedge funds tend to remain bullish in the face of uncertain markets. To a large extent, their jobs depend on remaining positive and securing further investments. These leaders of the investment world need to return to basics. Wall Street needs to again use a company's "book-value" as an analytical tool . A book-value reflects some natural reality that can bring share prices back down to earth. Using Enron as an example, I will explain.  In the second-quarter of 2001, Enron's book value was $13 a share, but, excluding goodwill, it was only $9 a share. Goodwill is a balance sheet entry on the asset side of the ledger, accepted when the investors overpaid when buying out another company.  At one stage Enron' shares were ten times this value! When a company is purchased for more than its book value, the excess is placed on the asset side of the ledger as goodwill. Although goodwill cannot be sold, it does inflate the price of the acquirer's book value! As such most reflected book values are in themselves an inflated reflection. In Enron's case, the book value was overstated by 44%! In the below table are some other book values (as of 12 November 2001), (Puetz, 2001).


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