Preliminary Remark about Pareto and Log-normal distributions.
It is custom in the economist to model the financial inequalities with the Pareto distribution (see e.g. https://en.wikipedia.org/wiki/Pareto_distribution ) which is essentially a polynomial function. The exact model of the inequalities is even worse and is closer to the log-normal distribution (see e.g. https://en.wikipedia.org/wiki/Log-normal_distribution ) where the severity of the inequalities is modeled with exponential functions.But here in this book we may keep talking about the Pareto distribution which is celebrated term among the economists
DIFFERENT FORMS OF THE LAW OF CORRESPONDENCE OR PARETO AMONG SCALES.
1) THE PARETO OR POWER RULE OF CYCLES OR TIME SCALES.
This is the basic rule among all scales and cycles , and it is a consequence of the law of inequalities or the universal attraction law in economy (see posts 25, 30 ). Roughly speaking it can be stated as follows
"More than 80% of the volumes of transactions are concentrated on less than 20% of the cycles, starting from the longest to the shortest"
As the volumes are responsible or the volatility and the amplitudes of the cycles, this law can be restated as follows.
"The amplitudes of the cycles, follow a power or Pareto distribution"
The longest cycles of 60 and 80 years, appear of course locally as stable trends. And in fact there is a non-cyclic stable artificial trend which is that of the indices, where the securities of enterprises enter when they are growing and when not are substituted by the securities of other enterprises. The Kuznet cycle of 22.2 years (or global climate solar cycle) is a medium one.
This law is also responsible to the fact the the longer term the trend (the larger the cycle) the less the "noise" of it , and thus the less the risk (As the "noise" can be considered on the spectral analysis, as the amplitudes of smaller cycles).
2) THE LOCAL APPROXIMATE N^(1/2) RULE
This rule applies when time scales are close enough and can be considered of the same cycle.
1) This n^(1/2) rule is an important rule as it calculates how the volatility and performance of trading systems changes when we shift time frames. It holds very well in the pure random walk, where it is an exact relation that if the volatility for a period p is s(p) and we want to know what is the volatility for period n*p , it is
It is custom in the economist to model the financial inequalities with the Pareto distribution (see e.g. https://en.wikipedia.org/wiki/Pareto_distribution ) which is essentially a polynomial function. The exact model of the inequalities is even worse and is closer to the log-normal distribution (see e.g. https://en.wikipedia.org/wiki/Log-normal_distribution ) where the severity of the inequalities is modeled with exponential functions.But here in this book we may keep talking about the Pareto distribution which is celebrated term among the economists
There are 3 contexts of laws required in trading . The appropriate LAWS OF THINKING for trading, the appropriate LAWS OF FEELINGS for trading , and the appropriate LAWS OF ACTIONS for trading.
The Successful trading is based according to these three laws on
1) POWER OF COLLECTIVE SCIENTIFIC THINKING: A GREAT AND SIMPLE SCIENTIFIC PERCEPTION OF THE FUNCTION OF THE ECONOMY THROUGH SOME GLOBAL STATISTICAL LAW. E.g. The law of Universal attraction in economy: that big money attracts more big money in the capital markets, and this by the balance of demand and supply makes securities indexes of the companies , that are indeed the big money, to have mainly stable ascending trend, whenever one can observe such one. Valid statistical deductions can be obtained with simple statistical hypotheses tests about the existence or not of a trend, with sample size half the period of a dominating cycle). (STABLE GREAT SCIENTIFIC THOUGHT-FORM OR BELIEF FACTOR IN TRADING. )
2) POWER OF COLLECTIVE PSYCHOLOGY: A LINK WITH THE POSITIVE COLLECTIVE PSYCHOLOGY.(E.g. that the growth of security indexes also represent the optimism of the growth and success of real business of the involved companies. And we bet or trade only on the ascension of the index, whenever an ascending trend is observable). (STABLE GREAT POSITIVE COLLECTIVE EMOTIONAL OR PSYCHOLOGICAL FACTOR IN TRADING. )
3) POWER OF INDIVIDUALS SIMPLE , CONSISTENT AND EASY TO CONDUCT PRACTICE. (e.g. a trading system with about 80% success rate that utilizes essentially only one indicator in 3 time frames, simple risk management rules of stop loss, take profit, trailing and escalation, and time spent not more than 20 minutes per day. In this way there are not many opportunities of human errors in the conduction of the trading practice. Failed trades are attributed to the randomness and are not to blame the trader). (STABLE SIMPLE AND EASY PRACTICAL FACTOR IN TRADING)
We may make the metaphor that successful trading is the ability to have successful resonance with the activities of top minority of those who determine the markets.
In trading there are 3 components in the feelings that must be dealt with. 1) The feeling of MONEY itself, 2) The feeling of the UTILITY of the money 3) The feeling of the RISK of the money each time. What is called usually money management in trading is essentially RISK MANAGEMENT.
VALID STATISTICS AND PREDICTABILITY
We must make here some remarks about the robust application of statistical predictions in the capital markets.
1) The theory that the efficient markets and in particular that they follow a pure random walk is easy to refute with better statistical experiments and hypotheses tests. The random walk would fit to a market where the sizes of the economic organizations are uniformly random. But the reality is that they follow a Pareto or power distribution, therefore this is inherited in the distribution of the volumes of transactions and also in the emerging trends or drifts.
2) The statistical models of time series are more robust , when they apply to the entity MARKET as a whole and are better as non-parametric , and not when they apply to single stocks and are linear or parametric. The reasons is that a time series as a stochastic process , requires data of a sample of paths, and for a single stock is available only a single path. While for all the market the path of each stock or security is considered one path from the sample of all paths of all the stocks. Linear time series models or derived like ARMA, ARIMA, SARIMA etc are destined to fail for particular patterns like those described in the post 32, because the true equations are non-linear and in addition with time varying coefficients!
3) The less ambitious the statistical application the more valid the result. E.g. applying a statistical hypothesis test, or analysis of variance to test if there is an up or a down trend (drift) or none, is a more valid statistical deduction , than applying a linear model of a time series and requiring prediction of the next step price.
4) Multivariate statistics, like factor analysis, discriminant analysis , logistic regression, cluster analysis , goal programming e.t.c., are possible to utilize for a more detailed theory of predictability and of portfolio analysis, and sector analysis of the market and not only H. Markowitz theory.
5) In applying of the above applications of statistics, the researcher must have at first a very good "feeling" of the data, and should verify rather with statistics the result rather than discover it.
6) The "Pareto rule of complexity-results" also holds here. In other words with less than 20% of the complexity of the calculations is derived more than 80% of the deduction. The rest of the 20% requires more than 80% more complexity in the calculations.
The Successful trading is based according to these three laws on
1) POWER OF COLLECTIVE SCIENTIFIC THINKING: A GREAT AND SIMPLE SCIENTIFIC PERCEPTION OF THE FUNCTION OF THE ECONOMY THROUGH SOME GLOBAL STATISTICAL LAW. E.g. The law of Universal attraction in economy: that big money attracts more big money in the capital markets, and this by the balance of demand and supply makes securities indexes of the companies , that are indeed the big money, to have mainly stable ascending trend, whenever one can observe such one. Valid statistical deductions can be obtained with simple statistical hypotheses tests about the existence or not of a trend, with sample size half the period of a dominating cycle). (STABLE GREAT SCIENTIFIC THOUGHT-FORM OR BELIEF FACTOR IN TRADING. )
2) POWER OF COLLECTIVE PSYCHOLOGY: A LINK WITH THE POSITIVE COLLECTIVE PSYCHOLOGY.(E.g. that the growth of security indexes also represent the optimism of the growth and success of real business of the involved companies. And we bet or trade only on the ascension of the index, whenever an ascending trend is observable). (STABLE GREAT POSITIVE COLLECTIVE EMOTIONAL OR PSYCHOLOGICAL FACTOR IN TRADING. )
3) POWER OF INDIVIDUALS SIMPLE , CONSISTENT AND EASY TO CONDUCT PRACTICE. (e.g. a trading system with about 80% success rate that utilizes essentially only one indicator in 3 time frames, simple risk management rules of stop loss, take profit, trailing and escalation, and time spent not more than 20 minutes per day. In this way there are not many opportunities of human errors in the conduction of the trading practice. Failed trades are attributed to the randomness and are not to blame the trader). (STABLE SIMPLE AND EASY PRACTICAL FACTOR IN TRADING)
We may make the metaphor that successful trading is the ability to have successful resonance with the activities of top minority of those who determine the markets.
In trading there are 3 components in the feelings that must be dealt with. 1) The feeling of MONEY itself, 2) The feeling of the UTILITY of the money 3) The feeling of the RISK of the money each time. What is called usually money management in trading is essentially RISK MANAGEMENT.
VALID STATISTICS AND PREDICTABILITY
We must make here some remarks about the robust application of statistical predictions in the capital markets.
1) The theory that the efficient markets and in particular that they follow a pure random walk is easy to refute with better statistical experiments and hypotheses tests. The random walk would fit to a market where the sizes of the economic organizations are uniformly random. But the reality is that they follow a Pareto or power distribution, therefore this is inherited in the distribution of the volumes of transactions and also in the emerging trends or drifts.
2) The statistical models of time series are more robust , when they apply to the entity MARKET as a whole and are better as non-parametric , and not when they apply to single stocks and are linear or parametric. The reasons is that a time series as a stochastic process , requires data of a sample of paths, and for a single stock is available only a single path. While for all the market the path of each stock or security is considered one path from the sample of all paths of all the stocks. Linear time series models or derived like ARMA, ARIMA, SARIMA etc are destined to fail for particular patterns like those described in the post 32, because the true equations are non-linear and in addition with time varying coefficients!
3) The less ambitious the statistical application the more valid the result. E.g. applying a statistical hypothesis test, or analysis of variance to test if there is an up or a down trend (drift) or none, is a more valid statistical deduction , than applying a linear model of a time series and requiring prediction of the next step price.
4) Multivariate statistics, like factor analysis, discriminant analysis , logistic regression, cluster analysis , goal programming e.t.c., are possible to utilize for a more detailed theory of predictability and of portfolio analysis, and sector analysis of the market and not only H. Markowitz theory.
5) In applying of the above applications of statistics, the researcher must have at first a very good "feeling" of the data, and should verify rather with statistics the result rather than discover it.
6) The "Pareto rule of complexity-results" also holds here. In other words with less than 20% of the complexity of the calculations is derived more than 80% of the deduction. The rest of the 20% requires more than 80% more complexity in the calculations.
DIFFERENT FORMS OF THE LAW OF CORRESPONDENCE OR PARETO AMONG SCALES.
1) THE PARETO OR POWER RULE OF CYCLES OR TIME SCALES.
This is the basic rule among all scales and cycles , and it is a consequence of the law of inequalities or the universal attraction law in economy (see posts 25, 30 ). Roughly speaking it can be stated as follows
"More than 80% of the volumes of transactions are concentrated on less than 20% of the cycles, starting from the longest to the shortest"
As the volumes are responsible or the volatility and the amplitudes of the cycles, this law can be restated as follows.
"The amplitudes of the cycles, follow a power or Pareto distribution"
The longest cycles of 60 and 80 years, appear of course locally as stable trends. And in fact there is a non-cyclic stable artificial trend which is that of the indices, where the securities of enterprises enter when they are growing and when not are substituted by the securities of other enterprises. The Kuznet cycle of 22.2 years (or global climate solar cycle) is a medium one.
This law is also responsible to the fact the the longer term the trend (the larger the cycle) the less the "noise" of it , and thus the less the risk (As the "noise" can be considered on the spectral analysis, as the amplitudes of smaller cycles).
2) THE LOCAL APPROXIMATE N^(1/2) RULE
This rule applies when time scales are close enough and can be considered of the same cycle.
1) This n^(1/2) rule is an important rule as it calculates how the volatility and performance of trading systems changes when we shift time frames. It holds very well in the pure random walk, where it is an exact relation that if the volatility for a period p is s(p) and we want to know what is the volatility for period n*p , it is
s(n*p)=(n^(1/2))*p. Where n is an integer or inverse of an integer. In other words when we multiply or divide the period by n, the volatility is multiplied or divided by n^(1/2).
This is an exact rule for the continuous time model (ITO stochastic calculus) of constant rate of return R and constant standard deviation σ of it. It is the model of the price movement of the securities, that the portfolio theory assumes. But it is not an exact rule if we use discrete time models!
2) This n^(1/2) rule is also the rule of the standard deviation of the sample mean (law of large numbers) which is s/(n^1/2) (see http://en.wikipedia.org/wiki/Sample_mean_and_sample_covariance ). Here it is supposed that the average rate of return R or growth rate of the trend is estimated as average in large scale, and the sample that it is used to estimate it as average R (of percentage changes of the prices per some period) has sample variance s/(n^1/2). That is, for very large sample the variance of the rate of the trend R is zero, while for smaller samples and thus time periods, it is s/(n^1/2). The smaller the time period the larger the sample variance of the rate R. The maximum is at one-element sample which is s.
3) Some times there is a n^(1/2) rule the state this law as follows.
X% of the time (e.g. X%=80%) you have stationary markets with neutral volatility and only (1-X)% of the time (e.g. (1-X)%=20%) the market is trending. Now this is X is larger at small scale, and shorter at large scale. The rule that the time of neutral stationary increases is by a rule of n^(1/2).
This is an exact rule for the continuous time model (ITO stochastic calculus) of constant rate of return R and constant standard deviation σ of it. It is the model of the price movement of the securities, that the portfolio theory assumes. But it is not an exact rule if we use discrete time models!
2) This n^(1/2) rule is also the rule of the standard deviation of the sample mean (law of large numbers) which is s/(n^1/2) (see http://en.wikipedia.org/wiki/Sample_mean_and_sample_covariance ). Here it is supposed that the average rate of return R or growth rate of the trend is estimated as average in large scale, and the sample that it is used to estimate it as average R (of percentage changes of the prices per some period) has sample variance s/(n^1/2). That is, for very large sample the variance of the rate of the trend R is zero, while for smaller samples and thus time periods, it is s/(n^1/2). The smaller the time period the larger the sample variance of the rate R. The maximum is at one-element sample which is s.
3) Some times there is a n^(1/2) rule the state this law as follows.
X% of the time (e.g. X%=80%) you have stationary markets with neutral volatility and only (1-X)% of the time (e.g. (1-X)%=20%) the market is trending. Now this is X is larger at small scale, and shorter at large scale. The rule that the time of neutral stationary increases is by a rule of n^(1/2).
4) This law has also a different formulation which deduces that the relative volatility per time unit is lower at larger time scales and larger in shorter time scales. (Seemingly converse of the previous rule on random walks) Every one that has observed a 50 years annual graph say of the SnP500, and a 50 minutes graph of SnP500 can at glance understands that at the 50 year annual graph there is relatively stable trend (drift) with small relative volatility per year as percentage change, while at the 50 minutes graph the trend is usually not stable, while the relative volatility per minute as percentage of change is quite high.
This has as effect that while a trading system that is profitable for a sufficient larger horizon, say at 15 minutes bars, will continue to be profitable if applied to say weekly bars, the converse is not true. Many systems that are profitable at weekly or daily bars, are no longer profitable when applied at 15 minutes or 5 minutes bars. The reason is that because at daily bars the volatility is low , a relatively simple recipe will be profitable, but when shifting at 15 minutes bars due to higher volatility the original simplicity is not adequate a higher complexity is required. We may call this minimum complexity that a system requires to be profitable in almost all (rainbow) time frames as the "threshold minimum complexity of the universality"
5) The markets are not a random walk. Still this law is empirically correct. In my model of the markets, the rainbow stochastic process, where the total behavior is composed from behaviors at each rainbow frequency, I do use that the amplitude of the cyclic phenomena from rainbow frequency to rainbow frequency, flows this law of n^(1/2).
6) As the results of many trading systems depend essentially on the volatility this empirical law holds also for the best possible performance of trading systems. E.g. Let us say that by studying the 50 years performance of commodex system (see e.g. http://commodex.com/ )which is applied on daily bars,we find that it has average return 110% and standard deviation of this average annual return 80%. Let us suppose that this commodex system has sufficient complexity to be profitable also in all (rainbow) time-frames. In other words that it has the "threshold minimum complexity of the universality". Le us assume further that being an algorithm manual or computerized that it is feasible to conduct is also over 15 minutes bars. What will be the average return and its volatility? The empirical law of n^(1/2) suggests the calculation average return over 15 minutes bars=(110%)*[(1day/15minutes)^(1/2)]=(110%)*[(1440/15)^(1/2)]=(110%)*[(96)^(1/2)]=1077.7% And its volatility would be (80%)*[(96)^(1/2)]=783%
These figure may seem very large. Of course all resulted rates of return of such a trading, if it were to be recorded by the bank accounting department and published, would be figures divided by 100, as the usual maximum nominal leverage in forex is 100. So all this effort would be equivalent to achieve what is the usual annual return of the markets (10%-12%) but with 100 times less variance of the equities curve, so that nominal maximum draw down would not exceed the 0.33%. In these calculations after the law of correspondence, the law of systematic progressive asymmetry among scales ,other many important practical factors are not taken in to account (e.g. increased noise-to-signal ratio as we go to smaller time scales thus less good predictability of the trend etc) . If all such factors are taken in to account, then depending on the particular system, the rate and its volatility at 15 minutes bars, may be from quite less to the above figures, to even negative and losing values.
7) Through actual statistical measurements of the volatility of the markets across the time frames, and comparison wit the curve n^(1/2) it is found that there is an increasing deviation from this n^(1/2) toward the 10 minutes to 1 minute bars.
A simple way to plot this curve and verify this n^(1/2) rule , is to estimate the average High-Low of bars of all the time scales e.g. minutes, M1, 5-minutes M5, 15 minutes M15, 30-minutes M30, one hour H1, 4-hours H4, 1-day D1, 1-week W1, 1-month MN1, and then plot this curve as number of minutes and average high-low which is almost a n^(1/2) curve.
See also wikipedia reference
http://en.wikipedia.org/wiki/Volatility_(finance)
8) One can take advantage of this law of the markets, together with the law of Pareto distribution of the draw-downs and draw-ups in a chart (see e.g. post 25 ) to create profitable grid trading, at the fastest possible time scale (thus High Frequency Trading HFT). This grid trading with its grid-pyramiding will take advantage of the Pareto law of micro-trends, while a tight take profit and large stop will take advantage of the difference of the volatility at short time scale (where is is higher, thus more often positions will open and close at the take profit) and the volatility at the larger time scale (where is is less).
A simple way to plot this curve and verify this n^(1/2) rule , is to estimate the average High-Low of bars of all the time scales e.g. minutes, M1, 5-minutes M5, 15 minutes M15, 30-minutes M30, one hour H1, 4-hours H4, 1-day D1, 1-week W1, 1-month MN1, and then plot this curve as number of minutes and average high-low which is almost a n^(1/2) curve.
See also wikipedia reference
http://en.wikipedia.org/wiki/Volatility_(finance)
8) One can take advantage of this law of the markets, together with the law of Pareto distribution of the draw-downs and draw-ups in a chart (see e.g. post 25 ) to create profitable grid trading, at the fastest possible time scale (thus High Frequency Trading HFT). This grid trading with its grid-pyramiding will take advantage of the Pareto law of micro-trends, while a tight take profit and large stop will take advantage of the difference of the volatility at short time scale (where is is higher, thus more often positions will open and close at the take profit) and the volatility at the larger time scale (where is is less).
9) Most of the traders think that intraday manual trading is radically more profitable than day-to-day trading where only once per day (e.g. for 15 minutes) a control and a decision is taken. They think so because they estimate that profits will increase in direct analogy to the shorter time scale they will use. But it is not so at least for two reasons. a) The shorter the scale the more the “noise” (=non profitable and non-tradable fluctuations of the markets due to unpredictability) b) In intraday manual trading, one has to spend many hours in front of the screen as if he was working at office while in day-to-day trading only 15-20 minutes per day, while he may have a normal work and normal day with other non-trading activities. c) Intraday and short time scales waves and patterns depend on a small number of people (mainly some packets of transactions by employees of the big banks) and therefore are subject to unpredicted changes, of the actions of these employees. But long term waves depend on a very larger volumes , and global populations involved in the economy, therefore are more stable.
The truth is that for manual trading the golden scale is the seasonal cycles (2-6 months) as sub-cycles of a 5.55 Kitchin cycle or 22.2 years global climate cycle (Kuznet cycle). Of course by programming automate trading it is possible to trade intra-day without spending human time. But the rate of return of such automated intraday trading is not higher than the seasonal manual trading if in the latter, the human pattern recognition is involved which is superior to automated trading pattern recognition. Some sellers of automated trading show the results of their programs for a limited time intervals (some months only) which appear very high so as to sell or rent them. But sooner or latter such automated trading had also significant failures so that in the long run (5 years or more) have less rate of return that the manual seasonal trading.
The way that unpredictability increases as the time scale becomes shorter, seems to be a result of the law of inequality. The very shape of the Pareto distribution could be used to present how unpredictability increases when the time sale decreases. In the next diagram of the Pareto didstribution, the x-axis is the time scale, and the y-axis is the unpredictability. A hint of why this is so is the next: A wave in shorter time scale takes less volumes of transactions to be shaped. And the volumes of transactions, follow the pareto or power distribution. The less the volumes the smaler the population (of transactions but also of traders) the less the predictability and stability.
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