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Unformatted text preview: Information, entropy and evolutionary finance Jing Chen
School of Business
University of Northern British Columbia
Prince George, BC
Canada V2N 4Z9
Web: http://web.unbc.ca/~chenj/ Abstract:
In economic literature, the cost of obtaining information is generally
determined exogenously. In physics, however, there is a long tradition to
link the entropic cost of obtaining information to the value of information.
Recently, an entropy theory of value, or information theory of value was
developed to provide a unified understanding of economic value,
information and physical entropy. In this work, we show how the entropy
theory of information and value provides clear understanding on the
evolving nature of investment strategies and on the long time debate on the
efficiency of financial markets. JEL classification: D80, G14
Keywords: Information, entropy, evolutionary finance, market efficiency Information, entropy and evolutionary finance 1. Introduction Financial activities are intrinsically linked to information. In economic
literature, the cost of obtaining information is generally determined
exogenously. (Grossman and Stiglitz, 1980) In physics, however, there is a
long tradition to link the physical cost of obtaining information to the value
of information. (Maxwell, 1871) After Shannon (1948) developed the
entropy theory of information, the relation between information and physical
entropy became explicit. (Bennett, 1988) Recently, an entropy theory of
value, or information theory of value was developed to provide a unified
understanding of economic value, information and physical entropy. (Chen,
2002) In this work, we show how the entropy theory of information and
value provides clear understanding on the evolving nature of investment
strategies and on the long time debate on the efficiency of financial markets.
We also show that the evolution of economic processes and investment
strategies is highly parallel to the evolution of biological systems. 1 There are two basic questions about the financial markets: how efficient are
the markets and will the markets become more efficient over time. Financial
markets are often thought to be highly efficient because of the low cost of
information. From the information theory, the value of information is
inversely related to probability. The information that can be obtained at low
cost and hence widely available is of low value. The information of high
value is usually carefully guarded and difficult to detect. Animals have
discovered this long ago. “In those cases where animal signals really are of
mutual benefit, they will tend to sink to the level of a conspiratorial whisper:
indeed this may often have happened, the resulting signals being to
inconspicuous for us to have noticed them. If signal strength increases over
the generations this suggests, on the other hand, that there has been
increasing resistance on the side of the receiver.” (Dawkins, 1999, p. 59)
Dawkins’ description of animal behavior can be applied to financial market
equally well. Once we recognize the high cost of obtaining information with
high value, the problem about the efficiency of financial market is very
similar to the problem of efficiency of real markets. (Farmer and Lo, 1999)
The real markets are generally thought to be less efficient because of the
high cost of business projects. However functional financial markets can
distribute the cost among many investors. 2 Most of the research efforts in financial markets are to reduce information
asymmetry as investors. Will all research efforts improve the efficiency of
the overall financial markets over time? While the information asymmetry of
a mature company or industry may reduce over time, the information
asymmetry of the whole financial market may not. The utilization of natural
resources generally follows from easy to difficult among living organisms.
(Atkins, 1995) Similarly, the development of human industries generally
follows from easy to difficult. As the technologies of mature industries
become widely known and less valuable, competition pressure drives
innovations and new industries. At time passes by, the new industries are
generally more and more difficult to develop and more and more difficult for
most people to understand. So information asymmetry will generally grow
instead of decrease. Recently, the stock prices of many of the high tech
companies dropped over ninety percent in a short period of time, signaling
the difficulty of the general investment public to understand the intricacy of
the complex technological systems. A fundamental result in information theory is that the amount of information
one can receive is the amount of information sent minus equivocation, which 3 is determined by statistical relation between senders and receivers of
information. (Shannon, 1948) When the correlation between the sender and
the receiver is zero, no information can be transmitted from the sender to the
receiver. When the correlation between the sender and the receiver is low,
very little information can be transmitted. When a company develops a new
product, a new organizational structure, a new marketing strategy, because
of the high level of equivocation, its value will only be gradually understood
by the public. Competitors will not rush in to drive down profit margin and
investors will not rush in to push the share prices to the level that reflected
the expected future profits. The high return on the shares of this type of
companies will persist over a prolonged period, as opposed to the instant
adjustment from efficient market theory. There are many ways to assess the validity of two different theories. One is
to see how the results can be extended to similar fields. The financial market
is often said to adjust to new information instantly because of intense
competition. Since the intellectual market is highly competitive as well, one
would expect it will adjust to new information in several years. However, it
is well known that theories of fundamental importance are often neglected
for many decades. (Khun, 1970) For example, Bachelier’s theory had to 4 wait more than half century before it got the attention. This is hard to
reconcile with efficient market theory. However, it is logical from
information theory. If a research result is an addition to a well established
theory, it will be noticed and accepted immediately because of low level of
equivocation. When a fundamentally new theory appears, however, the level
of equivocation is high and the theory may take great effort to get accepted.
The promotion of a fundamentally new idea is in general so difficult that
Wallace, the cofounder of the theory of evolution, gave much more credit to
the promotion of new ideas over their creation. “No one deserves either
praise or blame for the ideas that comes to him. … But the actions which
result from our ideas may properly be so treated, because it is only by patient
thought and work that new ideas, if good and true, become adapted and
utilized.” (George, 1964, p. 280) Another way to assess the validity of a theory on share prices of firms is to
examine its consistency with theories on other aspects of firms. From the
efficient market theory, a company’s stock have high return because of some
unforeseeable events that cannot be predicted. However, from the researches
in business strategy, a company does well often because it persists in a good
strategy for a long time before the competitors and the stock market react. 5 (Collins and Porras, 1994)Take Wal-Mart as an example. One of its most
important strategies is to set up large discount stores in small communities.
The early entry of one large store in a small community preempts the entry
of other big stores. The resulting local monopoly ensures high level of profit.
Since the value of information is positively related to scarcity, a player
adopting a superior strategy will keep quiet about it. To keep a low profile,
Wal-Mart avoided opening new stores where Sears and K-mart already had
existence. This gave other giant retailers the impression that Wal-Mart was
not very competitive. Hence other retailers will be less likely to imitate the
strategies of Wal-Mart. In fact, the strategy of local monopoly in small rural
communities was not copied by other giant retailers such as Sears and Kmart
for a long time for they thought small communities are too small markets for
big players. The extensive time lag in adopting a superior strategy from a
competitor is not consistent with efficient market theory, but is a natural
result from information theory. This paper is organized as follows. Section 2 briefly describes the history of
information theory and presents the information theory of value. Section 3
describes the product life cycle and explains how the evolution of economic
processes is often accompanied by evolution of investment strategies. 6 Section 4 discusses the consistence problem about the efficient market
theory and information theory with empirical evidences. Section 5
concludes. 2. Information theory and the information theory of value In a thought experiment, Maxwell (1871) reasoned, if information is
costless, the entropy of a system can be decreased, which violated the
second law of thermodynamics. He concluded that the physical cost of
obtaining information must be at least as much as the value of information.
After Shannon (1948) identified information as the reduction of entropy
mathematically, the equivalence between information and physical entropy
became explicit. (Bennett, 1988) The success of information theory stimulated many research in economics.
(Theil, 1967) However, some conceptual difficulties prevented the
development of an entropy theory of value, or “information theory of value”.
(Georgescu-Roegen, 1971; Arrow, 1999) A major difficulty is the prevalent
belief that information, once created, is not scarce in the economic sense.
This belief ignores a fundamental result in information theory: that the 7 information received is equal to information sent minus equivocation. There
is always a cost in overcoming equivocation. For example, it takes about
twenty years of education, with extremely high cost, before one can read
papers in many academic journals. Two persons trained in different
disciplines often have great difficulty communicating with each other. Once
we recognize the high cost of information processing, the entropy theory of
value or information theory of value becomes very natural. The value of information is generally determined exogenously in economic
literature due to the lack of an analytical theory. Recently, an entropy theory
of value, or information theory of value was developed to provide a unified
understanding of economic value, information and physical entropy. (Chen,
2002) The following is a brief introduction of the theory. The value of information is a function of probability. It satisfies the
following properties: 1. The information value of two events is higher than the value of
each of them. 8 2. If two events are independent, the information value of the two
events will be the sum of the two. 3. The information value of any event is non-negative. The only mathematical functions that satisfy all the above properties are of
the form H ( P ) = − log b P (1) where b is a positive constant. (Applebaum, 1996) Economic value is a function of scarcity. Scarcity can be defined as a
probability measure P in a certain probability space. It is generally agreed
that value of products satisfies the following properties: 1. The value of two products should be higher than the value of each of
them. 9 2. If two products are independent, that is, if two products are not
substitutes or partial substitutes of each other, then the total value of
the two products will be the sum of the two. 3. The value of any product is non-negative. Hence both information and economic value are represented by the same
function mathematically. Suppose a random event, X, has n discrete states, x1, x2, …,xn, each with
probability p1, p2, …,pn. The information value of X is the average of
information value of each state, that is n H ( X ) = −∑ p j log( p j ) (2) j =1 The right hand side of formula (2), which is the entropy function first
introduced by Boltzmann in 1870s, is the general form for information and
economic value. 10 Why both information and economic value are the reduction of entropy?
From the entropy law, the most universal law of the nature, the increase of
entropy of a system is spontaneous. The reduction of entropy in a system,
however, takes effort, which is the base for both information and economic
value. There are some differences between physical cost and economic cost. (Chen,
2002) Take the production of paper for example. Paper is made from wood.
The economic cost of paper production is the cost of cutting, transportation
and processing of wood and the cost of selling paper. All processes that
incur economic costs have physical costs. The physical cost also includes
the growth of wood, Which is ultimately provided by sunlight free of
economic cost. Figure 1 is a graph of formula (1), where H is a function of P, the probability
of event. From (1), we can see that value is a decreasing function of
probability. When P = 1, -log P =0. The value of information that is known
to everyone is zero. When P approaches zero, -logP approaches infinity. The
value of information that is known to few is very high. For example, if an
unexpected surge of corporate profit is known to very few people, i.e., when 11 P is very small and –log P is very big, the information is highly valuable.
Huge profit can be made by trading the underlying stocks. But when it is
known to many people, the value of information is very low. In general,
when some knowledge is mastered by many people, its market value is very
low. Because of the high value of information that is only known or understood
by few, important information is usually whispered in a small circle instead
of being broadcasted to the public. For example, the general public was
taken by surprise by the sudden collapse of Enron, although the frauds that
led to its eventual collapse had been going on for many years. All the parties
that involved in the frauds, including the auditing firm and the investment
banks, had an interest not to spread the news. Even if information is distributed freely, a receiver may not be able to
comprehend its full meaning. Following Shannon (1948), the amount of
information one can receive, R, would be obtained by subtracting from the
amount of information sent the average rate of conditional entropy. 12 R = H ( x) − H y ( x ) (3) The conditional entropy Hy(x) is called the equivocation. It measures the
average ambiguity of the received signal. (Shannon, 1948) When x and y are
independent, Hy(x) = H(x) and R = 0. No information can be transmitted
between two objects who are independent to each other. This shows that it is
very difficult for most people to understand the value of a new idea, product
or organizational structure. When x and y are linearly related, Hy(x) = 0 and
no information is lost in transmission. Since no two persons are linearly
related, whenever information is transmitted between people, part of it is lost
in the process. In general, when the correlation between the sender and the
receiver is low, very little information can be transmitted between them.
That the efficiency of information transmission depends on background
knowledge is well understood in many disciplines, such as culture studies.
(Hall, 1977) 13 3. Evolution of economic processes and evolutionary finance In this section, we will apply information theory and the information theory
of value to understand the evolution of economic processes and investment
strategies. First, we discuss the value change during product life cycles.
Suppose an economic system has M persons. The percentage of people who
have the product is p. Then the unit value of a product is
− log p (4) The total value of a product is Mp(− log p) (5) Figure 2 is the graph of unit value and total value of a product with respect
to its abundance. When a product is new and scarce, the unit value is high.
Its total value is low. As the production increases, the total value will
increase as the unit value decreases. When the production quantity is over a
certain level, however, the total value of a product will start to decrease as 14 well. Intuitively, this is easy to understand. The market values of
manufacturers of mature products are generally low. The evolution of industry life cycles are often accompanied by the evolution
of investment strategies. For companies in a new industry or adopting new
production systems, the equivocation about them is high. The high level of
information asymmetry protects the companies from intense competition and
prevents the investment public to make accurate estimation about their
values. If the high level of profits persists and share holders make high
return over a prolonged period, it will attract more research attention from
both competitors and investors. Information asymmetry will decrease over
time. As the company or industry specific information becomes widely
understood, the value of the information, which is inversely related to
probability, becomes very small. Once a successful investment or trading
strategy becomes well known, its value approaches zero. Investors need to
find new places for higher return. Will the evolution of economic processes and investment processes leads to
more efficient markets over time? While the information asymmetry about a
particular industry or strategy may reduce, the overall information 15 asymmetry of the financial markets may not reduce for the following
reasons: First, the development of human industries generally follows from easy to
difficult. At time passes by, the new industries are generally more and more
difficult to develop. The specialized technologies are more and more
difficult for most people to understand. So information asymmetry will
generally grow instead of decrease. This is reflected from the fact that the
transaction costs in economic activities has been growing over time. (Wallis
and North, 1986) Malkiel (1995) reported that while there was some persistence in superior performance of fund managers in the seventies, this
persistence disappeared in the eighties. This is usually interpreted as the
increasing efficiency of the market. However it can also be caused by that
the information asymmetry to mutual fund managers increases over time that
it is harder and harder for fund managers to gain insight about the dynamics
of the industry developments. Given tremendous wealth accumulated from
entrepreneur activities in the eighties and nineties, the increase of
information asymmetry probably offer an explanation that is closer to
reality. 16 Second, although mature companies in stable industries have already been
heavily studied, the emergence of new industries or new organizational
structures, which are not well understood by the investment public, may
seriously affect the value of mature companies. For example, the emergence
of Wal-Mart greatly affect the value of Sears, Kmart and other established
retailers. Even if one does a lot of research on Sears, she will not be able to
value Sears accurately if she does not understand the growth dynamics of
Wal-Mart. There is a similar pattern in biological evolution. The emergence
of new species often leads to the extinction of exisiting species. (Morowitz,
1992) 4. On empirical tests Theories are ultimately accepted or rejected from empirical evidences. At
the same time, theories often influence what empirical tests are performed
and how empirical tests are interpreted. This makes it difficult to establish
the validity of a theory from empirical tests. It is often more fruitful to
examine the consistency of a theory with theories concerning other aspects
of the same problems. 17 Empirical tests show that most mutual fund managers cannot outperform
markets. (Carhart, 1997) Since mutual fund managers are highly paid
professionals, their inability to outperform markets seem to suggest that the
markets are highly efficient. However, mutual funds managers cover a
spectrum of companies. Their knowledge of a particular company is
generally not as detailed as those company insiders. Since information
asymmetry is difficult to overcome, it is expected from the information
theory that most professional investors could not earn higher returns over the
market average. Both efficient market theory and information theory can
offer reasonable explanation on the inability of most mutual fund managers
to outperform. From the information theory, the difficulty of reduction of information
asymmetry enables a small minority of industry pioneers to make higher
than average return investing in publicly listed firms over prolonged period.
With the rare exception of Warren Buffett, most people who make high
return on investing publicly traded companies are people working in
particular industries. Even Warren Buffett himself is a manager of the
businesses he acquires. Since these people have much more intimate
knowledge about the industry than others, they generally have better 18 understanding of the industry dynamics and have lower level of ambiguity in
interpreting industry specific information. Many of these people, such as
Sam Walton and Bill Gates, hold the stocks of publicly traded companies
that offer significant higher rates than the market average over long periods.
If the markets are efficient, then investments in these publicly traded
companies should not offer higher rates of return than the market average.
From efficient market theory, the good performance of a particular stock is
caused by unpredictable random events. At the same time, the managers of
these best performing companies are often credited for their foresightedness. Both efficient market theory and information theory suggest that there are
only small numbers of businesses are highly inefficiently valued. However,
the assessment of the importance is very different. From the efficient market
theory, the less efficiently assessed firms are firms neglected by researchers
that are not very important to the overall economy. From the information
theory, the firms that have new visions about future or new ways of doing
business are difficult to value because of high degree of equivocation. These
firms, such as Wal-Mart, Microsoft, Dell, although small in number, often
have profound influences on the overall economy and the value of many
other businesses. This is the same in biological world. A small number of 19 species, such as human beings, have profound influence on the overall
biosphere and the fate of many other species. 5. Conclusion The efficient market theory has been under increasing scrutiny in recent
years. (Shefrin, 2000) In this work, we show that the entropy theory of
information provide clear understanding on the evolving nature of the
economic processes and investment strategies. “Once again animals discover the trick first. … Some species (of butterflies) evolved to be
poisonous or distasteful, and warned their predators with gaudy colors. ...
But then some nonpoisonous butterflies copied the colors, too, enjoying the
protection while avoiding the expense of making themselves distasteful.
When the mimics become too plentiful, the colors no longer conveyed
information and no longer deterred the predators. The distasteful butterflies
evolved new colors, which were then mimicked by the palatable ones, and
so on.” (Pinker, 1997, p. 501) Financial market, as a small sample in the vast
biological system, faithfully reflects the general pattern of unending
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University of Chicago Press, Chicago and London). 23 Figure Captions Figure 1: Information value and probability
Figure 2: The unit value and total value of a product with respect to scarcity 24 5 4 value 3 2 1 0
0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 scarcity 25 8 7 6 5 unit value
total value 4 3 2 1 0
0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 scarcity 26 ...
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