From: John Conover <john@email.johncon.com>

Subject: Re: portfolio management

Date: Sat, 19 Sep 1998 22:16:55 -0700

John Conover writes: > > Can the portfolio growth be made optimal? Yes, for example, by buying > stocks on margin, (I'll show that margins are not a practical strategy > when building a stock portfolio,) using the exact same methods we have > been using all along-and it too can be optimized-ie., the portfolio > growth can be maximized, while at the same time minimizing risk > exposure to the margin calls. The tsstock program, (which is not part > of the tsinvest suite-it is part of the fractal.tar.zip suite in > http://www.johncon.com/ndustrix/archive/fractal.tar.gz,) can be used to do it. > The trick is to buy enough, (but not more or less,) of the initial > investment in a group of stocks that make up a portfolio on > margin. How much on margin? It turns out to be: > The tsstock program is for simulating the gains of a stock investment using Shannon probability. Bottom line, it is for programmed trading (PT) of stocks. The C sources are freely available as Open Source software on http://www.johncon.com/ndustrix/archive/fractal.tar.gz. A fragment, specific to this discussion, of the manual page is attached ... John -- John Conover, john@email.johncon.com, http://www.johncon.com/ DESCRIPTION This program is an investigation into whether a stock price time series could be modeled as a fractal Brownian motion time series, and, further, whether, a mechanical wagering strategy could be devised to optimize portfolio growth in the equity markets. Specifically, the paradigm is to establish an isomorphism between the fluctuations in a gambler's capital in the speculative unfair tossed coin game, as suggested in Schroeder, and speculative investment in the equity mar- kets. The advantage in doing this is that there is a large infrastructure in mathematics dedicated to the anal- ysis and optimization of parlor games, specifically, the unfair tossed coin game. See Schroeder reference. Currently, there is a repository of historical price time series for stocks available at, http://www.ai.mit.edu/stocks.html that contains the historical price time series of many hundreds of stocks. The stock's prices are by close of business day, and are updated daily. The stock price history files in the repository are avail- able via anonymous ftp, (ftp.ai.mit.edu,) and the programs tsfraction(1), tsrms(1), tsavg(1), and tsnormal(1) can be used to verify that, as a reasonable first approximation, stock prices can be represented as a fractional Brownian motion fractal, as suggested by Schroeder and Crownover. (Note the assumption that, as a first approximation, a stock's price time series can be generated by independent increments.) This would tend to imply that there is an isomorphism between the underlying mechanism that produces the fluctu- ations in speculative stock prices and the the mechanism that produces the fluctuations in a gambler's capital that is speculating on iterations of an unfair tossed coin. If this is a reasonably accurate approximation, then the underlying mechanism of a stock's price time series can be analyzed, (by "disassembling" the time series,) and a wagering strategy, similar to that of the optimal wagering strategy in the iterated unfair coin tossing game, can be formulated to optimize equity market portfolio growth. As a note in passing, it is an important and subtile point, that there are "operational" differences in wager- ing on the iterated unfair coin game, and wagering on a stock. Specifically, in the coin game, a fraction of the gambler's capital is wagered on the speculative outcome of the toss of the coin, and, depending on whether the toss of the coin resulted in a win, (or a loss,) the wager is added to the gambler's capital, (or subtracted from it,) respectively. However, in the speculative stock game, the gambler wagers on the anticipated FLUCTUATIONS of the stock's price, by purchasing the stock. The important dif- ference is that the stock gambler does not win or loose an amount that was equal to the stock's price, (which was equivalent to the wager in the iterated unfair coin game,) but only the fluctuations of the stock's price, ie., it is an important concept that a portfolio's value (which has an investment in a stock,) and the stock's price do not, necessarily, "track" each other. In some sense, wagering on a stock is NOT like a gambler wagering on the outcome of the toss of an unfair coin, but like wagering on the capital of the gambler that wagered on the outcome of the toss of an unfair coin. A very subtile difference, indeed. Note that the paradigm of the isomorphism between wagering on a stock and wagering in an unfair tossed coin game is that the graph, (ie., time series,) of the gambler's capi- tal, who is wagering on the iterated outcomes of an unfair tossed coin, and the graph of a stock's price over time are statistically similar. If this is the case, at least in principle, it should be possible to "dissect" the time series of both "games," and determine the underlying statistical mechanism of both. Further, it should be, at least in principle, possible to optimize portfolio growth of speculative investments in the equity markets using information-theoretic entropic techniques. See Kelly, Pierce, Reza, and Schroeder. Under these assumptions, the amount of capital won or lost in each iteration of the unfair tossed coin game would be: V(t) - V(t - 1) for all data points in the gambler's capital time series. This would correspond to the amount of money won or lost on each share of stock at each interval in the stock price time series. Likewise, the normalized increments of the gambler's capi- tal time series can be obtained by subtracting the value of the gambler's capital in the last interval from the value of the gambler's capital in the current interval, and dividing by the value of the gambler's capital in the last interval: V(t) - V(t - 1) --------------- V(t - 1) for all data points in the gambler's capital time series. This would correspond to the fraction of the gambler's capital that was won or lost on each iteration of the game, or, alternatively, the fraction that the stock price increased or decreased in each interval. The normalized increments are a very useful "tool" in ana- lyzing time series data. In the case of the unfair coin tossing game, the normalized increments are a "graph," (or time series,) of the fraction of the capital that was won or lost, every iteration of the game. Obviously, in the unfair coin game, to win or lose, a wager had to be made, and the graph of the absolute value, or more appropri- ately, the root mean square, (the absolute value of the normalized increments, when averaged, is related to the root mean square of the increments by a constant. If the normalized increments are a fixed increment, the constant is unity. If the normalized increments have a Gaussian distribution, the constant is ~0.8 depending on the accu- racy of of "fit" to a Gaussian distribution,) of the nor- malized increments is the fraction of the capital that was wagered on each iteration of the game. As suggested in Schroeder, if an unfair coin has a chance, P, of coming up heads, (winning) and a chance 1 - P, of coming up tails, (loosing,) then the optimal wagering strategy would be to wager a fraction, f, of the gambler's capital, on every iteration of the game, that is: f = 2P - 1 This would optimize the exponential growth of the gam- bler's capital. Wagering more than this value would result in less capital growth, and wagering less than this value would result in less capital growth, over time. The vari- able f is also equal to the root mean square of the nor- malized increments, rms, and the average, avg, of the nor- malized increments is the constant of the average exponen- tial growth of the gambler's capital: t C(t) = (1 + avg) where C(t) is the gambler's capital. It can be shown that the formula for the probability, P, as a function of avg and rms is: avg --- + 1 rms P = ------- 2 where the empirical measurement of avg and rms are: n ----- 1 \ V(t) - V(t - 1) avg = - > --------------- n / V(t - 1) ----- i = 0 and, n ----- 2 2 1 \ [ V(t) - V(t - 1) ] rms = - > [ --------------- ] n / [ V(t - 1) ] ----- i = 0 respectively, (additionally, note that these formulas can be used to produce the running average and running root mean square, ie., they will work "on the fly.") The formula for the probability, P, will be true whether the game is played optimally, or not, ie., the game we are "dissecting," may not be played with f = 2P - 1. However, the formula for the probability, P: rms + 1 P' = ------- 2 will be the same as P, only if the game is played opti- mally, (which, also, is applicable in "on the fly" method- ologies.) Interestingly, the measurement, perhaps dynamically, (ie., "on the fly,") of the average and root mean square of the normalized increments is all that is necessary to optimize the "play of the game." Note that if P' is smaller than P, then we need to increase rms, by increasing f, and, like- wise, if P' is larger than P, we need to decrease f. Thus, without knowing any of the underlying mechanism of the game, we can formulate a methodology for an optimal wager- ing strategy. (The only assumption being that the capital can be represented as an independent increment fractal- and, this too can, and should, be verified with meticulous application of fractal analysis using the programs tsfrac- tion(1), tsrms(1), tsavg(1), and tsnormal(1).) At this point, it would seem that the optimal wagering strategy and analytical methodology used to optimize the growth of the gambler's capital in the the unfair tossed coin gain is well in hand. Unfortunately, when applying the methodology to the equity markets, one finds that, for almost all stocks, P is greater than P', perhaps tending to imply that in the equity markets, stocks are over priced. To illustrate a simple stock wagering strategy, suppose that analytical measurements are made on a stock's price time series, and it is found, conveniently, that P = P', implying that f = rms, (after computing the normalized increments of the stock's price time series and calculat- ing avg, rms, P, and P'.) Note that in the optimized unfair coin tossing game, that wagering a fraction, f = rms, of the gambler's capital would optimize the exponen- tial growth of the gambler's capital, and that the fluctu- ations, over time, of the gambler's capital would simply be the normalized increments of the gambler's capital. The root mean square of the fluctuations, over time, are the fraction of that the gambler's capital wagered, over time. To achieve an optimal strategy when wagering on a stock, the objective would be that the normalized increments in the value of the portfolio, and the root mean square value of the normalized increments of the portfolio, also, sat- isfy the criteria, f = rms. Note that the fraction of the portfolio that is invested in the stock will have normal- ized increments that have a root mean square value that are the same as the root mean square value of the normal- ized increments of the stock. The issue is to determine the fraction of the stock port- folio that should be invested in the stock such that that fraction of the portfolio would be equivalent to the gam- bler wagering a fraction of the capital on a coin toss. It is important to note that the optimized wagering strategy used by the gambler, when wagering on the outcome of a coin toss, is to never wager the entire capital, but to hold some capital in reserve, and wager only a fraction of the capital-and in the optimum case this wager fraction is f = rms. In a stock portfolio, even though the investment is totally in stocks, it could be considered that some of this value is wagered, and the rest held in reserve. The amount wagered would be the root mean square of the nor- malized increments of the stocks price, and the amount held in reserve would be the remainder of the portfolio's value. (Note the paradigm-there is an isomorphism between the fluctuating gambler's capital in the unfair coin toss- ing game, and the fluctuating value of a stock portfolio.) In the simple case where P = P', the fraction of the port- folio value that should be invested in the stock is f = root mean square of the stock's normalized increments, which would be the same as f = 2P - 1, where P = ((avg/rms) + 1) / 2 or P = (rms + 1) / 2. Note that the fluctuations in the value of the portfolio do to the fluc- tuations in the stocks price would be statistically simi- lar to the fluctuations in the gambler's capital when playing the unfair coin tossing game. This also leads to a generality, where P and P' are not equal. If the root mean square of the normalized incre- ments of the stock price time series are too small, say by a factor of 2, then the fraction of the portfolio invested in the stock should be increased, by a factor of 2 (in this example.) This would make the root mean square of the fluctuations in the value of the portfolio the same as the the root mean square of the fluctuations in the gambler's capital under similar statistical circumstances, (albeit with twice as much of the portfolio's equivalent "cash reserves" tied up in the investment in the stock. To calculate the ratio by which the fraction of the port- folio invested in a stock must be increased: avg --- + 1 rms P = ------- 2 and, f = 2P - 1 = rms and letting the measured rms by rms , m avg m ---- + 1 rms avg m m f = 2P - 1 = 2 -------- - 1 = ---- = rms 2 rms m (Note that both of the values, avg and rms, are functions of the probability, P, but their ratio is not.) and letting F be the ratio by which the fraction of the portfolio invested in a stock must be increased to accom- modate P not being equal to P': avg rms m F = ---- = ---- rms 2 m rms m and multiplying both sides of the equation by f, to get the fraction of the portfolio that should be invested in the stock while accommodating P not being equal to P': 2 avg avg avg m m m F * f = ---- * ---- = ---- 2 rms 3 rms m rms m m which can be computed, dynamically, or "on the fly," and where avg and rms are the average and root mean square of the normalized increments of the stock's price time series, and assuming that the stock's price time series is composed of independent increments, and can be represented as a fractional Brownian motion fractal. Representing such an algorithm in pseudo code: 1) for each data point in the stock's price time series, find the, possibly running, normalized increment from the following equation: V(t) - V(t - 1) --------------- V(t - 1) 2) calculate the, possibly running, average of all normalized increments in the stock's price time series by the following equation: n ----- 1 \ V(t) - V(t - 1) avg = - > --------------- n / V(t - 1) ----- i = 0 3) calculate the, possibly running, root mean square of all normalized increments in the stock's price time series by the following equation: n ----- 2 2 1 \ [ V(t) - V(t - 1) ] rms = - > [ --------------- ] n / [ V(t - 1) ] ----- i = 0 4) calculate the, possibly running, fraction of the portfolio to be invested in the stock, F * f: 2 avg m F * f = ---- 3 rms m To reiterate what we have so far, consider a gambler, iterating a tossed unfair coin. The gambler's capital, over time, could be a represented as a Brownian fractal, on which measurements could be performed to optimize the gambler's wagering strategy. There is supporting evidence that stock prices can be "modeled" as a Brownian fractal, and it would seem reasonable that the optimization techniques that the gambler uses could be applied to stock portfolios. As an example, suppose that it is desired to invest in a stock. We would measure the average and root mean square of the normalized increments of the stock's price time series to determine a wagering strategy for investing in the stock. Suppose that the measurement yielded that the the the fraction of the capital to be invested, f, was 0.2, (ie., a Shannon probability of 0.6,) then we might invest the entire portfolio in the stock, and our portfolio would be modeled as 20% of the portfolio would be wagered at any time, and 80% would be considered as "cash reserves," even though the 80% is actually invested in the stock. Additionally, we have a metric methodology, requiring only the measurement of the average and root mean square of the increments of the stock price time series, to formulate optimal wagering strategies for investment in the stocks. The assumption is that the stock's price time series is composed of independent increments, and can be represented as a fractional Brown- ian motion fractal, both of which can be verified through a metric methodology. Note the isomorphism. Consider a gambler that goes to a casino, buys some chips, then plays many iterations of an unfair coin tossing game, and then cashes in the chips. Then consider investing in a stock, and some time later, selling the stock. If the Shannon probability of the time series of the unfair coin tossing game is the same as the time series of the stock's value, then both "games" would be statistically similar. In point of fact, if the toss of the unfair coin was replaced with whether the stock price movement was up or down, then the two time series would be identical. The implication is that stock values can be modeled by an unfair tossed coin. In point of fact, stock values are, generally, fractional Brownian motion in nature, implying that the day to day fluctuations in price can be modeled with a time sampled unfair tossed coin game. There is an implication with the model. It would appear that the "best" portfolio strategy would be to continually search the stock market exchanges for the stock that has the largest value of the quotient of the average and root mean square of the normalized increments of the stock's price time series, (ie., avg / rms,) and invest 100% of the portfolio in that single stock. This is in contention with the concept that a stock portfolio should be "diver- sified," although it is not clear that the prevailing con- cept of diversification has any scientific merit. To address the issue of diversification of stocks in a stock portfolio, consider the example where a gambler, tossing an unfair coin, makes a wager. If the coin has a 60% chance of coming up heads, then the gambler should wager 20% of the capital on hand on the next toss of the coin. The remaining 80% is kept as "cash reserves." It can be argued that the cash reserves are not being used to enhance the capital, so the gambler should play multiple games at once, investing all of the capital, investing 20% of the capital, in each of 5 games at once, (assuming that the coins used in each game have a probability of coming up heads 60% of the time-note that the fraction of capital invested in each game would be different for each game if the probabilities of the coins were different, but could be measured by calculating the avg /rms of each game.) Likewise, with the same reasoning, we would expect that stock portfolio management would entail measuring the quo- tient of the average and root mean square of the normal- ized increments of every stock's price time series, (ie., avg / rms,) choosing those stocks with the largest quo- tient, and investing a fraction of the portfolio that is equal to the this quotient. Note that with an avg / rms = 0.1, (corresponding to a Shannon probability of 0.55-which is "typical" for the better performing stocks on the New York Stock Exchange,) we would expect the portfolio to be diversified into 10 stocks, which seems consistent with the recommendations of those purporting diversification of portfolios. In reality, since most stocks in the United States exchanges, (at least,) seem to be "over priced," (ie., P larger than P',) it will take more capital than is available in the value of the portfolio to invest, opti- mally, in all of the stocks in the portfolio, (ie., the fraction of the portfolio that has to be invested in each stock, for optimal portfolio performance, will sum to greater than 100%.) The interpretation, I suppose, in the model, is that at least a portion of the investment in each stock would be on "margin," which is a relatively low risk investment, and, possibly, could be extended into a formal optimization of "buying stocks on the margin." The astute reader would note that the fractions of the portfolio invested in each stock was added linearly, when these values are really the root mean square of the nor- malized increments, implying that they should be added root mean square. The rationale in linear addition is that the Hurst Coefficient in the near term is near unity, and for the far term 0.5. (By definition, this is the charac- teristic of a Brownian motion fractal process.) Letting the Hurst Coefficient be H, then the method of summing multiple processes would be: H H H V = V + V + ... tot 1 2 so in the far term, the values would be added root mean square, and in the near term, linearly. Note that this is also a quantitative definition of the terms "near term" and "far term." Since the Hurst Coefficient plot is on a log-log scale, the demarcation between the two terms is where 1 - ln (t) = 0.5 * ln (t), or when ln (t) = 2, or t = 7.389... The important point is that the "root mean square formula" used varies with time. For the near term, H = 1, and linear addition is used. For the far term, a root mean square summation process is used. (Note, also, that a far term H of 0.5 is unique to Brownian motion fractals. In general, it can be different than 0.5. If it is larger than 0.5, then it is termed fractional Brownian motion, depending on who is doing the defining.) There are some interesting implications to this near term/far term interpretation. First, the "forecastability" is better in the near term than far term-which could be interpreted as meaning that short term strategies would yield better portfolio performance than long term strate- gies-see the Peters reference, pp. 83-84. Secondly, it can be used to optimize portfolio long term strategy. For example, suppose that a stock's Shannon probability is 0.52, and all stocks in the portfolio have the same Shan- non probability. This means that the portfolio should con- sist of 25 stocks. However, in the long run, the portfolio would have a root mean square value of the square root of 25 times 0.04, or 0.2. This would tend to imply that, on the average, over the long run, the stock portfolio would be one fifth of the total investments. Naturally, this ratio could be adjusted, over time, depending on the instantaneous value of the Shannon probabilities of all different investments, like bonds, metals, etc. This would imply that "timing of the market" would have to be initiated to adjust the ratio of investment in stocks. One of the implications of entropic theory is that this is impossible. However, as the Shannon probability of the various investments change, statistical estimation can be used to asses the statistical accuracy of these movements, and the ratios adjusted accordingly. This would tend to suggest that adaptive computational control system method- ology would be an applicable alternative. As a note in passing, the average and root mean square of the normalized increments of a stock's price time series, avg and rms, respectively, represent a qualitative metric of the stock. The average, avg, is an expression of the stock's growth in price, and the root mean square, rms, is a expression of the stock's price volatility. It would seem, incorrectly, at first glance that stocks should be selected that have high price growth, and low price volatility-however, it is a more complicated issue since avg and rms are interrelated, and not independent of each other. See the references for theoretical concepts. In the diversified portfolio, the "volatilities" of the individual stocks add root mean square to the volatility of the portfolio value, so, everything else being equal, we would expect that the volatility of the portfolio value to be about 1 /3 the volatility of the stocks that make up the portfolio. (The ratio 1 / came from square root of 1 / 10, which is about 1 / 3.) (There is a qualification here, it is assumed that all stock price time series are made up of independent increments, and can be represented as a fractional Brownian motion fractal-note that this state- ment is not true if the time series is characterized as simple Brownian motion, like the gambler's capital in the unfair coin toss game-see Schroeder, pp. 157 for details.) So, it can be supposed, if one desires maximum performance in a stock portfolio, then one should search the stock market exchanges for the stock that has the highest quo- tient of the average and root mean square of the normal- ized increments of stock price time series, and invest 100% of the portfolio in that stock. As an alternative strategy, one could diversify the portfolio, investing in multiple stocks, and lower the portfolio volatility at the expense of lower portfolio performance. Arguments can probably be made for both strategies.

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