From: John Conover <firstname.lastname@example.org>
Subject: Re: Financial engineering article
Date: Mon, 23 Dec 1996 16:44:48 -0800
Same as before, typographical corrections, footnote  wordsmithing. John BTW, if you look at the "El Faro" bar problem, (new footnote ,) it is kind of interesting. It is what is known as a game-theoretic prediction problem with a mixed strategy Nash equilibrium of 60%. It turns out that one should predict more than 60% occupancy with a probability of 0.4, and less than 60% occupancy with a probability of 0.6, and then just mix up your decisions with those probabilities in an adequately random fashion, so that no one else can predict what you are going to do. (If you think about it, this is an obvious solution.) That is the best long term strategy for the "El Faro" bar problem, (short of calling all hundred people and asking them what they are going to do-which is a bit of an inconvenience just to go have a beer every week.) If we assume that everyone is real smart, (or by chance, over time, "learns" how to play the "game,") then we would have a hundred random processes, each with a probability of 0.4 of not going. And if we sum all hundred of them together, we would have a fractal time series for the occupancy of the bar over time. Obviously, a fractal (of the random walk, or Brownian motion variety, in this case,) solution is stable, over time, and is self reinforcing, ie., we would expect the occupancy of El Faro's to be a fractal, forever. Here is why. Now assume that some, maybe even many, of the people wishing to go to the El Faro formulate new strategies in an attempt to better their predictive position. Some strategies will work, some of the time, but no strategy will work all of the time-ie., these attempts will become the random process itself-and, over time, will settle back to the Nash equilibrium. The very act of attempting to better their predictive position reinforces the fractal solution, and in fact becomes the "engine" of the fractal itself. Obviously, the concept is applicable to many economic scenarios-like the stock market, industrial markets, etc. -- John Conover, email@example.com, http://www.johncon.com/ If we consider capital, V, invested in a savings account, and calculate the growth of the capital over time: V(t) = V(t - 1)(1 + a(t))........................(1.1) where a(t) is the interest rate at time t, (usually a constant.) In equities, a(t) is not constant, and varies, perhaps being negative at certain times, (meaning that the value of the equity decreased.) This fluctuation in an equity's value can be represented by modifying a(t) in Equation 1.1: a(t) = f(t) * F(T)..............................(1.2) where the product f * F is the fluctuation in the equity's value at time t. An equity's value, over time, is similar to a simple tossed coin game [Sch91, pp. 128], where f(t) is the fraction of a gambler's capital wagered on a toss of the coin, at time t, and F(t) is a random variable, signifying whether the game was a win, or a loss, ie., whether the gambler's capital increased or decreased, and by how much. The amount the gambler's capital increased or decreased is f(t) * F(t). In general, F(t) is a function of a random variable, with an average, over time, of avgf, and a root mean square value, rmsf, of unity. Note that for simple, time invariant, compound interest, F(t) has an average and root mean square, both being unity, and f(t) is simply the interest rate, which is assumed to be constant. For a simple, single coin game, F(t) is a fixed increment, (ie., either +1 or -1,) random V(t) = V(t - 1)(1 + f(t) * F(t))................(1.3) and subtracting V(t - 1) from both sides: V(t) - V(t - 1) = V(t - 1) (1 + f(t) * F(t)) - V(t - 1)........................................(1.4) and dividing both sides by V(t - 1): V(t) - V(t - 1) --------------- = V(t - 1) V(t - 1) (1 + f(t) * F(t)) - V(t - 1) -------------------------------------...........(1.5) V(t - 1) and combining: V(t) - V(t - 1) --------------- = V(t - 1) (1 + f(t) * F(t) ) - 1 = f(t) * F(t)............(1.6) We now have a "prescription," or process, for calculating the characteristics of the random process that determines an equity's value. That process is, for each unit of time, subtract the value of the of the equity at the previous time from the value of the equity at the current time, and divide this by the value of the equity at the previous time. The root mean square of these values are the root mean square of the random process. The average of these values are the average of the random process, avgf. The root mean square of these values can be calculated by any convenient means, and will be represented by rms. The average of these values can be found by any convenient means, and will be represented by avg. Therefore, if f(t) = f, and does not vary over time: rms = f.........................................(1.7) which, if there are sufficiently many samples, is a metric of the equity value's "volatility," and: avg = f * F(t)..................................(1.8) and if there are sufficiently many samples, the average of F(t) is simply avgf, or: avg = f * avgf..................................(1.9) which is a metric on the equity value's rate of "growth." Note that this is the "effective" compound interest rate from Equation 1.1. Equations 1.7 and 1.9 are important equations, since they can be used in portfolio management. For example, Equation 1.7 states that the volatility of the capital invested in many equities, simultaneously, is calculated as the root mean square of the individual volatility of the equities. Equation 1.9 states that the growths in the same equity values add together linearly. Dividing Equation 1.9 by Equation 1.7 results in the two f's canceling, or: avg --- = avgf.....................................(1.10) rms There may be analytical advantages to "model" avg as a simple tossed coin game, (either played with a single coin, or multiple coins, ie., many coins played at one time, or a single coin played many times.) The number of wins minus the number of losses, in many iterations of a single coin tossing game would be: P - (1 - P) = 2P - 1...........................(1.11) where P is the probability of a win for the tossed coin. (This probability is traditionally termed, the "Shannon probability" of a win.) Note that from the definition of F(t) above, that P = avgf. For a fair coin, (ie., one that comes up with a win 50% of the time,) P = 0.5, and there is no advantage, in the long run, to playing the game. However, if P > 0.5, then the optimal fraction of capital wagered on each iteration of the single coin tossing game, f, would be 2P - 1. Note that if multiple coins were used for each iteration of the game, we would expect that the volatility of the gambler's capital to increase as the square root of the number of coins used, and the growth to increase linearly with the number of coins used, irregardless of whether many coins were tossed at once, or one coin was tossed many times, (ie., our random generator, F(t) would assume a binomial distribution and if the number of coins was very large, then F(t) would assume, essentially, a Gaussian distribution.) Many equities have a Gaussian distribution for the random process, F(t). It may be advantageous to determine the Shannon probability to analyze avg --- = avgf = 2P - 1............................(1.12) rms or: avg --- + 1 = 2P...................................(1.13) rms and: avg --- + 1 rms P = -------....................................(1.14) 2 where only the average and root mean square of the normalized increments need to be measured, using the "prescription" or process outlined above. Interestingly, what Equation 1.12 states is that the "best" equity investment is not, necessarily, the equity that has the largest average growth, avgf. The best equity investment is the equity that has the largest growth, while simultaneously having the smallest volatility. In point of fact, the optimal decision criteria is to choose the equity that has the largest ratio of growth to volatility, where the volatility is measured by computing the root mean square of the normalized increments, and the growth is computed by averaging the normalized increments. We now have a "first order prescription" that enables us to analyze fluctuations in equity values, although we have not explained why equity values fluctuate. For a formal presentation on the subject, see the bibliography in [Art95] which, also, offers non-mathematical insight into the explanation. Consider a very simple equity market, with only two people holding equities. Equity value "arbitration" (ie., how equity values are determined,) is handled by one person posting (to a bulletin board,) a willingness to sell a given number of stocks at a given price, to the other person. There is no other communication between the two people. If the other person buys the stock, then that is the value of the stock at that time. Obviously, the other person will not buy the stock if the price posted is too high-even if ownership of the stock is desired. For example, the other person could simply decide to wait in hopes that a favorable price will be offered in the future. So the stock seller must not post a price that the other person would consider too high, and the other person would not buy at the price if it is reasoned that the seller's pricing strategy will be to lower the offering price in the future, which would be a reasonable deduction if the posted price is considered too high. What this means is that the seller must consider not only the behavior of the other person, but what the other person thinks the seller's behavior will be, ie., the seller must base the pricing strategy on the seller's pricing strategy. Such convoluted logical processes are termed "self referential," and the implication is that the market can never operate in a consistent fashion that can be the subject of deductive analysis [Pen89, pp. 101]. As pointed out by [Art95, Abstract], these types of indeterminacies pervade economics. What the two players do, in absence of a deductively consistent and complete theory of the market, is to rely on inductive reasoning. They form subjective expectations or hypotheses about how the market operates. These expectations and hypothesis are constantly formulated and changed, in a world that forms from others' subjective expectations. What this means is that equity values will fluctuate as the expectations and hypothesis concerning the future of equity values change. The fluctuations created by these indeterminacies in the equity market are represented by the term f(t) * F(t) in Equation 1.3, and since there are many such indeterminacies, we would anticipate F(t) to have a Gaussian distribution. This is a rather interesting conclusion, since analyzing the actions of aggregately many "agents," each operating on subjective hypothesis in a market that is deductively indeterminate, can result in a system that can not only be analyzed, but optimized. The only remaining derivation is to show that the optimal wagering strategy is, as cited above: f = rms = 2P - 1...............................(1.15) where f is the fraction of a gambler's capital wagered on each toss of a coin that has a Shannon probability, P, of winning. Following [Rez94, pp. 450], consider that the gambler has a private wire into the future who places wagers on the outcomes of a game of chance. We assume that the side information which he receives has a probability, P, of being true, and of 1 - P, of being false. Let the original capital of gambler be V(0), and V(n) his capital after the n'th wager. Since the gambler is not certain that the side information is entirely reliable, he places only a fraction, f, of his capital on each wager. Thus, subsequent to n many wagers, assuming the independence of successive tips from the future, his capital is: w l V(n) = (1 + f) (1 - f) V (0).................(1.16) where w is the number of times he won, and l = n - w, the number of times he lost. These numbers are, in general, values taken by two random variables, denoted by W and L. According to the law of large numbers: 1 lim - W = P..........................(1.17) n -> infinity n 1 lim - L = q = 1 - P..................(1.18) n -> infinity n The problem with which the gambler is faced is the determination of f leading to the maximum of the average exponential rate of growth of his capital. That is, he wishes to maximize the value of: 1 V(n) G = lim - ln ----....................(1.19) n -> infinity n V(0) with respect to f, assuming a fixed original capital and specified P: W L G = lim - ln (1 + f) + - ln (1 - f)..(1.20) n -> infinity n n or: G = P ln (1 + f) + q ln (1 - f)................(1.21) which, by taking the derivative with respect to f, and equating to zero, can be shown to have a maxima when: dG P - 1 1 - P -- = P(1 + f) (1 - f) - df 1 - P - 1 (1 - P)(1 - f) (1 + f)P = 0...........(1.22) combining terms: P - 1 1 - P 0 = P(1 + f) (1 - f) - P P (1 - P)(1 - f) (1 + f ) ......................(1.23) and splitting: P - 1 1 - P P(1 + f) (1 - f) = P P (1 - P)(1 - f) (1 + f) .......................(1.24) then taking the logarithm of both sides: ln (P) + (P - 1) ln (1 + f) + (1 - P) ln (1 - f) = ln (1 - P) - P ln (1 - f) + P ln (1 + f).......(1.25) and combining terms: (P - 1) ln (1 + f) - P ln (1 + f) + (1 - P) ln (1 - f) + P ln (1 - f) = ln (1 - P) - ln (P)............................(1.26) or: ln (1 - f) - ln (1 + f) = ln (1 - P) - ln (P)...........................(1.27) and performing the logarithmic operations: 1 - f 1 - P ln ----- = ln -----............................(1.28) 1 + f P and exponentiating: 1 - f 1 - P ----- = -----..................................(1.29) 1 + f P which reduces to: P(1 - f) = (1 - P)(1 + f)......................(1.30) and expanding: P - Pf = 1 - Pf - P + f........................(1.31) or: P = 1 - P + f..................................(1.32) and, finally: f = 2P - 1.....................................(1.33) Footnotes:  For example, if a = 0.06, or 6%, then at the end of the first time interval the capital would have increased to 1.06 times its initial value. At the end of the second time interval it would be (1.06), and so on. What Equation 1.1 states is that the way to get the value, V in the next time interval is to multiply the current value by 1.06. Equation 1.1 is nothing more than a "prescription," or a process to make an exponential, or "compound interest" mechanism. In general, exponentials can always be constructed by multiplying the current value of the exponential by a constant, to get the next value, which in turn, would be multiplied by the same constant to get the next value, and so on. Equation 1.1 is nothing more than a construction of V (t) = exp(kt) where k = ln(1 + a). The advantage of representing exponentials by the "prescription" defined in Equation 1.1 is analytical expediency. For example, if you have data that is an exponential, the parameters, or constants, in Equation 1.1 can be determined by simply reversing the "prescription," ie., subtracting the previous value, (at time t - 1,) from the current value, and dividing by the previous value would give the exponentiating constant, (1 + at). This process of reversing the "prescription" is termed calculating the "normalized increments." (Increments are simply the difference between two values in the exponential, and normalized increments are this difference divided by the value of the exponential.) Naturally, since one usually has many data points over a time interval, the values can be averaged for better precision-there is a large mathematical infrastructure dedicated to precision enhancement, for example, least squares approximation to the normalized increments, and statistical estimation.  "Random variable" means that the process, F(t), is random in nature, ie., there is no possibility of determining what the next value will be. However, F can be analyzed using statistical methods [Fed88, pp. 163], [Sch91, pp. 128]. For example, F typically has a Gaussian distribution for equity values [Cro95, pp. 249], in which case the it is termed a "fractional Brownian motion," or simply a "fractal" process. In the case of a single tossed coin, it is termed "fixed increment fractal," "Brownian," or "random walk" process. In any case, determination of the statistical characteristics of Ft are the essence of analysis. Fortunately, there is a large mathematical infrastructure dedicated to the subject. For example, F could be verified as having a Gaussian distribution using Chi-Square techniques. Frequently, it is convenient, from an analytical standpoint, to "model" F using a mathematically simpler process [Sch91, pp. 128]. For example, multiple iterations of tossing a coin can be used to approximate a Gaussian distribution, since the distribution of many tosses of a coin is binomial-which if the number of tosses is sufficient will represent a Gaussian distribution to within any required precision [Sch91, pp. 144], [Fed88, pp. 154].  Equation 1.3 is interesting in many other respects. For example, adding a single term, m * V(t - 1), to the equation results in V(t) = v(t - 1) (1 + f(t) * F(t) + m * V(t - 1)) which is the "logistic," or 'S' curve equation,(formally termed the "discreet time quadratic equation,") and has been used successfully in many unrelated fields such as manufacturing operations, market and economic forecasting, and analyzing disease epidemics [Mod92, pp. 131]. There is continuing research into the application of an additional "non-linear" term in Equation 1.3 to model equity value non-linearities. Although there have been modest successes, to date, the successes have not proved to be exploitable in a systematic fashion [Pet91, pp. 133]. The reason for the interest is that the logistic equation can exhibit a wide variety of behaviors, among them, "chaotic." Interestingly, chaotic behavior is mechanistic, but not "long term" predictable into the future. A good example of such a system is the weather. It is an important concept that compound interest, the logistic function, and fractals are all closely related.  In this section, "root mean square" is used to mean the variance of the normalized increments. In Brownian motion fractals, this is computed by sigmatotal^2 = sigma1^2 + sigma2^2 ... However, in many fractals, the variances are not calculated by adding the squares, (ie., a power of 2,) of the values-the power may be "fractional," ie., 3 / 2 instead of 2, for example [Sch91, pp. 130], [Fed88, pp. 178]. However, as a first order approximation, the variances of the normalized increments of equity values can successfully be added root mean square [Cro95, kpp. 250]. The so called "Hurst" coefficient, which can be measured, determines the process to be used. The Hurst coefficient is range of the equity values over a time interval, divided by the standard deviation of the values over the interval, and its determination is commonly called "R / S" analysis. As pointed out in [Sch91, pp. 157] the errors committed in such simplified assumptions can be significant-however, for analysis of equities, squaring the variances seems to be a reasonable simplification.  For example, many calculators have averaging and root mean square functionality, as do many spreadsheet programs-additionally, there are computer source codes available for both. See the programs tsrms and tsavg. The method used is not consequential.  There are significant implications do to the fact that equity volatilities are calculated root mean square. For example, if capital is invested in N many equities, concurrently, then the volatility of the capital will be rms / sqrt (N) of an individual equity's volatility, rms, provided all the equites have similar statistical characteristics. But the growth in the capital will be unaffected, ie., it would be statistically similar to investing all the capital in only one equity. What this means is that capital, or portfolio, volatility can be minimized without effecting portfolio growth-ie., volatility risk can addressed. Further, it does not make any difference, as far as portfolio value growth is concerned, whether the individual equities are invested in concurrently, or serially, ie., if one invested in 10 different equities for 100 days, concurrently, or one could invest in only one equity, for 10 days, and then the next equity for the next 10 days, and so on. The capital growth would have the same characteristics for both agendas. (Note that the concurrent agenda is superior since the volatility of the capital will be the root mean square of the individual equity volatilities divided by the square root of the number of equities. In the serial agenda, the volatility of the capital will be simply the root mean square of the individual equity volatilities.) Almost all equity wagering strategies will consist of optimizing variations on combinations of serial and concurrent agendas. There are further applications. For example, Equation 1.6 could be modified by dividing both the normalized increments, and the square of the normalized increments by the daily trading volume. The quotient of the normalized increments divided by the trading volume is the instantaneous growth, avg, of the equity, on a per-share basis. Likewise, the square root of the square of the normalized increments divided by the daily trading volume is the instantaneous root mean square, rmsf, of the equity on a per-share basis, ie., its instantaneous volatility of the equity. (Note that these instantaneous values are the statistical characteristics of the equity on a per-share bases, similar to a coin toss, and not on time.) Additionally, it can be shown that the range-the maximum minus the minimum-of an equity's value over a time interval will increase with the square root of of the size of the interval of time [Fed88, pp. 178]. Also, it can be shown that the number of expected stock value "high and low" transitions scales with the square root of time, meaning that the probability of an equity value "high or low" exceeding a given time interval is proportional to the square root of the time interval [Schroder, pp. 153].  Here the "model" is to consider two black boxes, one with a stock "ticker" in it, and the other with a casino game of a tossed coin in it. One could then either invest in the equity, or, alternatively, invest in the tossed coin game by buying many casino chips, which constitutes the starting capital for the tossed coin game. Later, either the equity is sold, or the chips "cashed in." If the statistics of the equity value over time is similar to the statistics of the coin game's capital, over time, then there is no way to determine which box has the equity, or the tossed coin game. The advantage of this model is that gambling games, such as the tossed coin, have a large analytical infrastructure, which, if the two black boxes are statistically the same, can be used in the analysis of equities. The concept is that if the value of the equity, over time, is statistically similar to the coin game's capital, over time, then the analysis of the coin game can be used on equity values. Note that in the case of the equity, the terms in f(t) * F(t) can not be separated. In this case, f = rms is the fraction of the equity's value, at any time, that is "at risk," of being lost, ie., this is the portion of a equity's value that is to be "risk managed." This is usually addressed through probabilistic methods, as outlined below in the discussion of Shannon probabilities, where an optimal wagering strategy is determined. In the case of the tossed coin game, the optimal wagering strategy is to bet a fraction of the capital that is equal to f = rms = 2P - 1 [Sch91, pp. 128, 151], where P is the Shannon probability. In the case of the equity, since f = rms is not subject to manipulation, the strategy is to select equities that closely approximate this optimization, and the equity's value, over time, on the average, would increase in a similar fashion to the coin game. The growth of either investment would be equal to avg = rms^2, on average, for each iteration of the coin game, or time unit of equity investment. This is an interesting concept from risk management since it maximizes the gain in the capital, while, simultaneously, minimizing risk exposure to the capital.  Penrose, referencing Russell's paradox, presents a very good example of logical contradiction in a self-referential system. Consider a library of books. The librarian notes that some books in the library contain their titles, and some do not, and wants to add two index books to the library, labeled "A" and "B," respectively; the "A" book will contain the list of all of the titles of books in the library that contain their titles; and the "B" book will contain the list of all of the titles of the books in the library that do not contain their titles. Now, clearly, all book titles will go into either the "A" book, or the "B" book, respectively, depending on whether it contains its title, or not. Now, consider in which book, the "A" book or the "B" book, the title of the "B" book is going to be placed-no matter which book the title is placed, it will be contradictory with the rules. And, if you leave it out, the two books will be incomplete.)  [Art95] cites the "El Farol Bar" problem as an example. Assume one hundred people must decide independently each week whether go to the bar. The rule is that if a person predicts that more than, say, 60 will attend, it will be too crowded, and the person will stay home; if less than 60 is predicted, the person will go to the bar. As trivial as this seems, it destroys the possibility of long-run shared, rational expectations. If all believe few will go, then all will go, thus invalidating the expectations. And, if all believe many will go, then none will go, likewise invalidating those expectations. Predictions of how many will attend depend on others' predictions, and others' predictions of others' predictions. Once again, there is no rational means to arrive at deduced a-priori predictions. The important concept is that expectation formation is a self-referential process in systems involving many agents with incomplete information about the future behavior of the other agents. The problem of logically forming expectations then becomes ill-defined, and rational deduction, can not be consistent or complete. This indeterminacy of expectation-formation is by no means an anomaly within the real economy. On the contrary, it pervades all of economics and game theory [Art95].  Interestingly, the system described is a stable system, ie., if the players have a hypothesis that changing equity positions may be of benefit, then the equity values will fluctuate-a self fulfilling prophecy. Not all such systems are stable, however. Suppose that one or both players suddenly discover that equity values can be "timed," ie., there are certain times when equities can be purchased, and chances are that the equity values will increase in the very near future. This means that at certain times, the equites would have more value, which would soon be arbitrated away. Such a scenario would not be stable. Bibliography: [Art95] W. Brian Arthur. "Complexity in Economic and Financial Markets." Complexity, 1, pp. 20-25, 1995. Also available from http://www.santafe.edu/arthur, February 1995. [Cro95] Richard M. "Crownover. Introduction to Fractals and Chaos." Jones and Bartlett Publishers International, London, England, 1995. [Fed88] Jens Feder. "Fractals." Plenum Press, New York, New York, 1988. [Mod92] Theodore Modis. "Predictions." Simon & Schuster, New York, New York, 1992. [Pen89] Roger Penrose. "The Emperor's New Mind." Oxford University Press, New York, New York, 1989. [Pet91] Edgar E. Peters. "Chaos and Order in the Capital Markets." John Wiley & Sons, New York, New York, 1991. [Rez94] Fazlollah M. Reza. "An Introduction to Information Theory." Dover Publications, New York, New York, 1994. [Sch91] Manfred Schroeder. "Fractals, Chaos, Power Laws." W. H. Freeman and Company, New York, New York, 1991.