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.

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.

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.

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..

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.

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

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).

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.

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 variablesÂ 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.

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!

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.
Â Â Â 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.

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.Â 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).