14. RE-POSITIONING THE PORTFOLIO. The portfolio of opportunities: Fluctuations and multiplication of profit.

1) In classical buy-and-hold investments the utilization of a portfolio of investments is as it is know to reduce the variance-risk of the account equities. The problem of defining the percentages of the participation of each asset or instruments in the portfolio is an interesting challenge. A classical solution, after the Nobel prize  of Markovitz, can be found e.g.  in the book:  

ELTON E.J.-GRUBER M.J. (1991) Modern portfolio theory and investment analysis Wiley.

From this book I personally utilize the cut-off-rate algorithm that is based on the assumption of the single index market model, the forecasted for the future horizon of the portfolio of the average rates Ri  , they standard deviations Si , and their beta coefficients bi  relative to the overall index of the market (that can be a weighted average of all traded instruments). The beta coefficients have a direct relation and simple derivation from the matrix of correlations of all instruments. I have coded this algorithm to run and create and update portfolios for more than  300 instruments even in a minute to minute frequency.

Assuming that all n in number instruments are uncorrelated between them , of equal standard deviation and all beta are equal, then the algorithm gives a simple practical formula: If the forecasted return for an instruments is Ri, then is percentage in the portfolio must be ai=Ri/(R1+R2+...+Rn). In other words as its return contributes to  the sum of all returns

Nevertheless this algorithm is not a trading system in its own as it depends heavily on the forecast returns Ri, therefore it is only a part of a trading system. Assuming that we have a good forecasting system and after application of the cut-off-rate procedure, the adjustment of the size of the positions on the instruments so that the portfolio percentages are kept constant during the forward horizon, is a trading by itself. It is a trading that the fluctuations of the prices during the horizon are assumed unpredictable, and still their occurrence has a favorable effect on the profit, compared to the non-existence of fluctuations. See also on the category "speculators" in the post no 3.

2) An experienced modern trader on instruments of high leverage (commodities, forex etc) has already understood that the real goal of the tradings (e.g. in forex) is not to maximize profits, but to minimize the final variance of the equities curve in the account. Once this is done then by the utilization of the leverage and reinvestment this can lead to very high profits. The effect of a portfolio is that it decreases the variance of the average return by an order of n^(1/2) where n is the number of instruments in the portfolio,  Therefore the effect of using a portfolio (e.g. in commodities) is eventually the increase of the profits.

3) But again in modern trading there is another very strong reason that the use of portfolio increases dramatically profits. Most good trading systems do not have continuously open positions. They carefully observe the market, and open positions for small time intervals (e.g. 20% of the time), while most of the time (e.g. 80%) have not open positions. We may call this  trading systems with high intermittency. Then while the use of portfolio for  systems that are continuously in the market has as a first effect that the portfolio averages their returns and decreases the variance, the use of portfolio for systems of high intermittency is that it almost adds-up the rates of return of each system.  To give a simplistic example, let as say that the system A trades Monday-Wednesday-Friday and has return Ra, while the system B trades Tuesday-Thursday and has return Rb. Then the superposition or simultaneous run  of the two systems will have return Ra+Rb  and not (Ra+Rb)/2 . We may call this phenomenon "the packing effect of systems of high intermittency".

Summarizing we say that while for buy-and-holders the use of a portfolio reduces the variance risk, for traders the use of a portfolio increases in a fantastic way  the profits.

13. THE ONLY CERTAINTY IN TRADING WITH MARGIN. FIRST PRIORITY IS TO BOUND LOSSES WITH THE KELLY CRITERION.The exposure , the growth management and the withdraws constant ratio rule

THE ONLY CERTAINTY IN TRADING.  FIRST PRIORITY IS TO BOUND LOSSES WITH THE KELLY CRITERION.


The wrong exposure management and over-exposure is probably the number one cause of failure and margin call bankruptcy, to all the accounts of small initial funds. But this means it is the case to the majority of small individual traders. Huge funds accounts trading managers, are not so much threatened by the over-exposure, because as a rule (although it is non-optimal) they apply systematic under-exposure. But with small and very small initial funds accounts, systematic under-exposure is not something very practical.


Once I watched during 2002 a Russian Physics Scientist (that we were working in Brussels in the same University) now in Switzerland  to perform a manual and semi-automated very succesful intraday trading for 100 days online. He doubled 100 K $ in 100 days. His maximum exposure rule was simple: To reserve for each contract so much funds that it becames of leverage=1, in other words without leverage. He started with under-exposure compared to the exposure corresponding to Leverage=1 and increased it gradually till the maximum exposure corresponding to leverage=1. It is probably not an exact optimal rule but it is for sure a safe tactic.


We can call it the rule of Leverage=1


Alexander Elder in his books and many other older and celebrated traders recommend a fixed exposure of 2% of the available funds. We may call it the rule of 2 %


It seems that for Alexander Elder it might have been his "holy grail" that transformed his losing trading in to winning trading. And indeed just correcting the over-exposure is enough to make the trading from losing in to profitable for a majority of traders.


Ralph Vince in his book "The mathematics of Money Management"  explains that the optimal exposure is to be found by computer iterative optimization based only on the information of all the trades their gain or loss, and their percentage of succesful-unsuccesful when trading with fixed position size. In other words it is a variable parameter depending on the system's backtest. He furthermore suggest a composite empirical formula, including the optimal f, the margin and the maximum draw-dawn of the system's backtest.


Other experts (like Keith Fitshen) insist that using the maximum drawdown is an oversestimation of the risk, as the maximum drawdown will increase indefinetly as the backtest horizon increases. He suggests using the
average starting or absolute draw-down and the standard deviation of the average draw-down.
It can even be derived a formula for the average starting or absolute draw down under general hypotheses of the Equity curve growth. (See post 33, the formula is s^2/|r| , where s^2 is the variance of the equity curve, (without reinestment money mamagement) and r is the drift (slope) of the equity curve.In fact the histogram of the absolute draw downs follows the exponential distribution, see http://en.wikipedia.org/wiki/Exponential_distribution)
 So he suggest estimating an accumulative histogram of draw-downs to calculate all that. His remarks are certainly correct, amd his methods, better from a statistical point of view. This is what I am using myself. I always calculate the histogram of draw-downs on any backtest, or historic performace of any system. After calculating the accumulative histogram of draw-downs we may apply a value at risk analysis and estimate what is the frequency or probability of a particular draw down. In addition what is the probability of crashing the account if a particular exposure is folowed, after this particular histogram of draw-downs. As a rule while starting with little funds we tend to apply under-exposure, compared to the optimal exposure; if we survive, and have accumulated significant ammount of gained funds, we may increase the exposure but still systematic under-exposure compared to the optimal exposure. Besides the larger size of the funds makes a small percentage of profits, a significant absolute ammount of money, adequate for normal consumtion needs.


Other traders prefer a fixed exposure (e.g. a rule of 6%) and then apply it to trade by trade, calculating the exposure based on the stop loss of each trade.


There are therefore mathematical-statistical laws of optimal exposure.


There is also subjective "feeling comfortable" exposure management conduction.


In general the final result is as smart as the dumpiest trade. (Ralph Vince)


Multiplying a large profit with zero makes always zero. (Warren Buffett)


"While reinvesting can turn a profitable system in to losing one,no reinvesting can turn a losing system in to profitable system." (Ralph Vince)


Focusing to make every single trade the best quality trade possible, it can always work, while it will not work if you focus to make every single trade profitable. So in conduction quality goes first and money afterwards.


Over-exposure above the optimal exposure will lead with almost certainty to crashing the account. Under-exposure falling in arithmetic progression, leads average return falling with geometric progression.

Once we have a trading system good enough to be successful with trades where StopLoss<=TakeProfit we may apply a reinvestment technique based on the Kelly formula:

f=(bp-(1-p))/b=p-(1-p)/b  . where f=optimal percentage of funds to risk, b=TakeProfit/StopLoss (it has to be >=1) and p=probability of success of the trade. (see http://en.wikipedia.org/wiki/Kelly_criterion)

For example let us say that we have an account of 1000$, and our trading system has a general success rate of 66% (after back-tests with constant lot size). Let us assume that we take a signal for a trade, that its stop-loss SL=20 and take-profit TP=30 have ratio TP/SL=1.5 

Then the Kelly formula can be used to define the lot size: The optimal percentage to risk is f=(1.5*0.66-0.33)/1.5=0.44 or 44%. In other words we may risk 440$, and for the 20 pips stop loss this is converted in to 440/20=22$/pip or 2.2 standard lots! Seems quite a big number indeed.In practice we should trade with smaller lot size, (e.g. a fixed percentage of what the kelly formula gives; my experients show that 5%-25% of the Kelly percentage is a safer choice) because the Kelly formula is derived with the assumption of continous subdivision of the funds and constant success rate of the trades, while in practice, the minimum lot size,and rather variable winning rate of the trades, does not permit such a procedure.If we continue to apply this formula, if the account grows, larger, more lots are opened so it is both a de-investment (when the results go bad) and re-investment (when the results go well) money management system. 

We may also derive directly the optimal leverage from the above Kelly formula and the SL, as
OptimalLeverage=((f*Equity)/SL*10)*100=(((((TP/SL)*p-(1-p))/TP/SL)*Equity)/SL*10)*100 or
 OptimalLeverage=(((((TP/SL)*p-(1-p))/TP/SL)*Equity)/SL*10)*100
In practice, my backtests show that one should take a constant smaller percentage over the Kelly variable  percentage.

Upon this money management system we may superimpose the optimal adjustment management system, described in another post that somehow does the reverse.

Instead of analysing further the exposure management I recommend reading the book by Ralph Vince : The Mathematics of Money Management.

The previous considerations lead to a withdrwal rule that e.g. has been applied for many years by professor Michael LeBoeuf in his investments (see https://en.wikipedia.org/wiki/Michael_LeBoeuf and http://www.nightingale.com/beat-time-money-trap-mp3.html )

 CONSTANT RATIO WITHDRAWAL RULE . We may divide the funds to 2/3 of them that we trade, and 1/3 that we do not trade. The exact percentage should be defined by the ratio (f=R/a^2) (that we mention above from the book  "Stochastic Differential equations" by B. K. Oksendal (Springer editions) page 223, example 11.5)  where R is the average per period rate o return of the trading on the used funds, measured on a sample of periods and a^2 is the variance of this rate of return on this sample of  periods.  E.g. of the rate of return is 10% per period and the standard deviation a of it is 34%, then the percentage f is 2/3. Each period we re-adjust the total funds so that this ratio f applies as division of the funds. As the sample of measurement of this ratio is not small, usually it remains rather constant. We withdraw e.g. from the 1/3 non-trading funds ,each period never more than  half of the average profits per period of the other 2/3 of the funds that are traded. 

12. The law of Pareto or correpsondence among scales: The systematic deviation from exact self-similarity

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

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.

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).  

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).

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. 

10. THE R.W. BABSON METHOD OF TRADING BY MEDIANS AND 1/3 OF THE PRICE MOVES. The power law of volumes and the law of financial action ( the Newtons laws of action) .

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.The statistical quantities from the front-office in trading need to me measured are the price position in the channel around the average, the velocity (1st derivative) and the acceleration-deceleration (2nd derivative), which is done as statistical quantities by a hypothesis test or confidence interval. The support-resistance levels can be measured also by action-volume histograms.  The measurements are done with convenient indicators, and can also define in a statistically valid way, not only , the channels , the trend, reversal, and Eliot-waves but also the spikes. In addition for the back-office of trading we need to measure the probability of success of trade based on the past history of trades, to apply the Kelly criterion, and also the average rate of increase of the trading funds and its variance again from the past history of trading.   (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 random, time varying coefficients that derive the random emergence of the 4 basic observable patterns (see post 32 ). In addition the standard application of the time series by the researchers,  focuses  on stationary time series after they extract  a stable exponential trend, while in the reality the main concern should be the random path of the average value of the prices that shapes the patterns and is neither constant exponential trend neither zero ! The "statistical momentum conservation" might then be nothing else than an hypothesis that the random and time varying 1st order in time steps , partial correlation of the prices , is always positive. This can be easily tested statistically. E.g. in the cross exchange rate EURUSD but also in the indexes, the partial correlation of the current to the previous time step bar is measured indeed positive, in almost all time frames, except at the daily time frame, where the cyclic behavior prevails. In the daily time frame the partial correlation is negative , which means if one day is up the next day it is more probable that it is down. In addition, the cyclic behavior is even stronger in pairs of two days with negative partial correlation (two days up two days down etc). In searching for random cycles or periodicity, of say a single index or even instrument , the valid statistical practice requires the creation of a sample of paths over a time interval of  a whole period, by collecting  the pieces of the path at different periods as the market move as far as the searched periodicity is concerned may be considered as moving independently at independent periods. 

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 , conjoint analysis, correspondence analysis, multidimensional scaling etc , goal programming etc are possible to utilize for a more detailed theory of 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 less hypothesis we use in applying statistical hypotheses, the better. That is why non-parametric statistics is better. An exception is our knowledge of the application of the Pareto distribution in various aspects of the market which we is parametric. 

That is why we avoid applying very complicated with many hypotheses and time consuming to estimate models to forecast the markets, but we prefer to respond to the market, by measuring only in a valid statistical way, the average position of the price, and the channel around it, the velocity (trend, 1st derivative) and acceleration-deceleration (2nd derivative)  of the prices. 

The statistical quantities from the front-office in trading need to me measured are 
1) the price position in the channel around the average, 2) the velocity (1st derivative) and 
3) the acceleration-deceleration (2nd derivative), which is done as statistical quantities by a hypothesis test or confidence interval. 
4) The support-resistance levels can be measured also by action-volume histograms.  The measurements are done with convenient indicators, and can also define in a statistically valid way, not only , the channels , the trend, reversal, and
5)  (Eliot) waves but also 
6) the spikes. 
7) It is required also an in advance in the past measurement and discovery of the basic stable cycles in the markets (see post 5)
8) An in advanced in the past measurement and discovery that trends duration and length, and volumes follow the Pareto distribution (see post 11,25 etc). 
In addition for the back-office of trading we need to measure the 
9) probability of success of trade based on the past history of trades, to apply the Kelly criterion, and also 
10) the average rate of increase of the trading funds and
11)  its variance again from the past history of trading.

The back-office statistical quantities in 9), 10), 11) are related , by simulation as e.g. in the simulator in post 43.

I have also created a simulator to experiment with different rules of withdrawals. 


There are many who complain that the indicators have lag, and prefer not to use indicators at all, but only the prices. This is rather stupid! The indicators when are measuring a statistical quantity  MUST have lag, because we are not interested for the price at the now only, but in a short-term past horizon too, which defines the statistical momentum which is statically conserved. It is not a race of speed, it is  a challenge of successful perception. The science of statistics is the best for the moment one can have , from the collective scientific thinking, and collective consensus, and we must  be honest and humble to admit is restricted abilities, but also trust , respect it and be confident for the success   when applying it. When we are applying the statistical mode of thinking for the markets, we never run serious dangers of being "burned"and "busted" in our deductions, as statistics claims everything only up to some probability, or probability inequality and intervals.  

There are some also that claim to have "intuitive guessing" about how the markets will move beyond the observable state of the markets. This of course cannot be included easily in  the standard statistical inference. But it seems to me that sometimes, this is in certain human and social environmental conditions is too much to ask from yourself, and it may fire-back to systematic opposite to the actual markets moves guessing! I believe that fortunately pure statistical inference from the observable states of the market only,  may be adequate for very profitable trading.   

Next we state intuitively the Laws of action of Newton for the markets, or the law of statistical momentum conservation as result of demand-supply polarity (see also post 22 ) and the Pareto law which is derived from the inequalities in the markets. 




0) The Power law of Volumes.

The distribution of the volumes, in a given granulation of time, follows a power or Pareto distribution. If take the logarithm of the sizes then the distribution becomes a straight line. This low follows and is an inherited property in the volumes of transactions, from the  well known and proved law of power or Pareto distribution of the sizes of (financial or nations) organizations. As a consequence more than 80% of the volume transactions are made from less than 20% of the organizations. 

1) The law of momentum conservation.

This law is somehow called in trading, "the law of trend following". On this conservation of the trend is based the success oof the trend following, as when a trend is detected, and we open positions we can have profit, only if the trend continuous for some time, which the trend conservation. Here the rule of volume enters too. The volume corresponds to the inertia mass (which is variable therefore), and as the momentum is measured as the product of speed and inetria, so the "momentum" of an instrument is estimated as the product of the price change during a bar (Close-Open),with the volume of transactions during the bar.  The reason that such a low exists in the financial  markets is closely related to the previous 0th power law of the volumes or  law of power or Pareto , which follows from the well known and proved low of power or Pareto distribution of the sizes of (financial or nations) organizations. If take the logarithm of the sizes then the distribution becomes a straight line. As a consequence more than 80% of the volume transactions are made from less than 20% of the organizations. At each time moment only organization is the winner (this holds even if two such maximal size organizations, are coupled in a domination mode interaction under some Voltera equations, see below)).  Thus when a very large winner organization decides according to their intentions to make a "transaction" , it will be so large, that it has to be executed in to a sequence of smaller packets, in a time interval, that will create a temporarily sustained conserved momentum or trend. 

The volumes measurement do provide better forecasting. The true rules of volumes are
1. A momentum acceleration is a true acceleration, if the volumes are increasing too.
2. A momentum deceleration is a true deceleration, if the volumes are decreasing too. 
Ofcourse the converse does not hold: An acceleration can be true, even if the volumes are not increasing. But if they are increasing we are sure it is true acceleration. The same with the deceleration.

2) The law of force and acceleration
While the law of the conservation of the momentum of financial activities is essential to have the ability of profit, in practice it is not the trend that has to be detected, as most people think but rather  the acceleration-deceleration of it.

In the financial markets, the "force" is through imbalance of  demand-supply and volume of orders and transactions.

To give a kinaesthetic key metaphor: Imagine your friend is driving the car with you inside, and you want a signal that he is going to turn 360 degrees backwards. If you wait to measure by the momentum of the car, it will always be after he has turned back and after some meters for the car to build opposite momentum. So it will be always late. While if you go by the acceleration as signal, you will feel the deceleration of the brakes quite before he turns the steering wheel. This simple realisation when converted to indicators is the basis for good anticipation (prediction, forecasting). It is like reading tomorrows newspaper.According to R.W. Babson it was what made him multi-millionaire. Present time traders use to term it as divergence in appropriate indicators.
Also according to the interpretation of volume as "inertia", the "force" of the coupling of the (non-Marhalian) demand-supply of populations, is best measured by the derivative (change) of the "momentum" (which involves also the volume) , and not only of the price.
Thus low volume times will have low inertia, thus big changes in the prices (high volaitlity) with small demand-supply forces.

3) The law of action-reaction

This law works through the law of polarity (see post 9) of demand supply, In other words that whenever there is a force of demand also a force of supply (and vice versa) is active in the markets. Also known in the slang of trading as Bulls-Bears. It was the favorite law for trading of Roger Ward Babson who was claiming that it was what made him multi-millionaire.

The action-reaction of Demand-Supply is in detailed described by the 3 modes of the coupling equations of Demand-Supply (see post 22 and 32).

For similar metaphor of laws from physics to social action, see also the interesting talk of the Google marketing manager on Physics and marketing at TED.com


http://www.ted.com/talks/dan_cobley_what_physics_taught_me_about_marketing.html


 In short the basic 3 modes of non-linear  interaction of populations or volumes  orders are


a) Win-Lose (Domination or pray-predator)


b) Lose-lose (Competition)


c) Win-Win (Cooperation)






You can even conveive 3 phases of the evolution of capitalism according to what


prevails statistically the a) Domination (Win-Lose) or Tyranny b) The


Competition (lose-lose) or the present state of capitalism c) The Cooperation


(Win-Win) or Meta-capitalism.






These 3 modes give rise to 4 patterns of price action (singularities)


a) Lighnings (or spikes)


b) Winds (or Trends in the  market)


c) Cyclons (or wavelets or technical analysis' triagles and rombs)


d) Calmness (flat noise or ranging market)



THE R.W. BABSON METHOD OF TRADING BY MEDIANS AND 1/3 OF THE PRICE MOVES. 

Here is the link where you can read the Dr Alan H. Andrew method and Roger Ward Babson method to implement the present law of Newtonian  momentum conservation and action-reaction through equality of  2 last successive waves (median lines method or pitchfork) . This method predict the next wave (from the previous) as well as the next reaction (or retrace). It is also one more application of the principle that the prediction based on the smallest discrete step memory (here one last full period wave) has also the least prediction error (exactly as in ARMA(1,1) time series as in the philosophy of Box-Jenkins). In the absence of a longer time frame trend, the  action and reaction are equal.  It also is used to place both the take profit and stop loss which are both  tight and trailing with SL<=TP. In particular this method shows that "random" deviations of the exact rule, are propagated in the next prediction creating therefore the way that markets move in  "seemingly" random ways. This method has much similarity with the forecasting method of ARMA(1,1) time series except here it is not numeric or algebraic  but rather pictorial  geometric or vector space analytic. In this method the part ofthe price motion that is predicted is rather small , but probably quite certain and occuring very often! 

Roger Ward Babson (https://en.wikipedia.org/wiki/Roger_Babson) had correctly predicted the great depression crisis of 1929 with his method above. He was friend with I. Fisher, and both became ill with tuberculosis, a deadly disease at their time (like cancer nowadays). But both with their confidence and keen self-discipline through working in the open fresh air and healthy life, managed to cure themselves. R.W. Babson became multi-Millionaire and thought so much as his lucky concept the action-reaction Newtonian law, to predict the seemingly random fluctuations of the markets, that he bought the house of Sir I. Newton.

http://www.trading-naked.com/alan_andrews_course_1.htm


also


http://amgallo.com/trading/bisects.htm


and
http://www.medianline.com/


A book:
http://www.amazon.com/Best-Trendline-Methods-Andrews-Techniques/dp/0965051838/ref=pd_bbs_1/002-2167389-1442403?ie=UTF8&s=books&qid=1194482162&sr=1-1


and another book


http://www.median-line-study.com/median-line-books.html#finding


and still another book
http://www.amazon.com/Trading-Median-Lines-Mapping-Markets/dp/0972982906/ref=sr_1_3?ie=UTF8&s=books&qid=1309594739&sr=1-3