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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 Phone: 1-250-960-6480 Fax: 1-250-960-5544 Email: chenj@unbc.ca 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 evolution. 20 References Applebaum, D. 1996, Probability and Information, an integrated approach. 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The Bell System Technical Journal. 27, 379-423, 623-656. 22 Shefrin, H., 2000, Beyond greed and fear : understanding behavioral finance and the psychology of investing, (Harvard Business School Press, Boston, Mass.). Theil, H., 1967, Economics and information theory. (North-Holland, Amsterdam) Wallis, J. and D. North, 1986, Measuring the transaction sector in the American economy, 1870-1970, in Long-term Factors in American Economic Growth, Edited by S. Engerman and R. Gallman. (The 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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