Scientific Perspective : Black Swan : Causality- Adjusted Randomized Factor Based Investing : Duality of Determinism & Randomness in Nature, Life & Markets VOL1

Scientific Perspective: Black Swan : Causality Adjusted Randomized Factor Investing : Determinism Inside Randomness

 

By

Pankaj Mani

(manipankaj9@gmail.com)

 

May 2023

 

 

1.Introduction : In this paper, I’ll discuss about the Scientific Perspectives of Financial Investments especially Factor Investing. First will talk about Philosophy & Development in Science and its connection with  Financial Markets. How Investment/Factor Investing can be made scientific Will discuss about some fundamental misconceptions of popular statistical tools e.g. Regression Analysis, Backtesting etc.  Also the Role of Time Dimension in Statistics.  Will  discuss about the Scientific Origin of Black Swan, Convexity and finally a New Causal approach of Factor based Investment.

Will also discuss that ultimate intellectual maturity is the acknowledgement of duality/two extremes in Nature in the form of Randomness(Uncertainty) & Determinism(Predictability) like Wave Particle Duality. Rather than being competitor to opposite being complementary to each other. It attempts to unify the two extremes of Black Swan Randomness and Causality in the dual form.

 

Note : I’m not going to make it much technical in mathematics, for to be understood by all. Also, there exist Fundamental issues with the conventional statistical and mathematical tools, which is blindly applied in finance without understanding the concept. So, my focus is on concepts behind the mathematical/statistical tools,which are often misused and misunderstood in finance. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.Background of Science :

 

Science has evolved from Newton’s  Classical Laws to Quantum Mechanics over the time. Einstein’s Theory of Relativity to Schrodinger’s wave Equation to Heisenberg Uncertainty Principle to Principle of Least Action to Feynman Path Integral Approach  have been developed in the study of science over the time. Science is still trying to unify the two extremes of Classical & Quantum world laws. One one hand, things appear Deterministic in day to day Classical world, things appear highly Uncertain/Random/Probabilistic in Quantum world. Science is trying to understand the Quantum world of Uncertainty and Unifying it to the Classical world.

Albert Einstein throughout his life couldn’t digest that God is playing dice with us. He believed in Deterministic world inherently which has been refuted over the time. On contrary, Niel’s Bohr’s  Principle of Complementarity emphasizes the role of Observer(Human Brain) through which the External world is experienced. Even Stephen Hawking talked in context of theory of Everything in Science. This is infact true that Science can’t find the ultimate theory of Everything without knowing about Human Brains, Consciousness etc. There is also Paradox of Consciousness and I wish I could take this here to show how Science needs radical approach and try to unify the ultimate reality of the Universe- Brain mutuality. But this is beyond the scope of this paper here. Hence, I would like to confine here.

Feynman’s Path Integral approach has been the most important scientific result in Science which originated from Paul Dirac’s Principle of Least Action. I would state that even Classical world theories like Newton’s Laws of Motion etc originate from the Quantum Energy Laws.

Those who have studied Physics know that one can derive Newton’s Laws of Motion from Principal of Least Action which is also supposed to be the core of Quantum world Laws. It states that any system tries to trace the path of Least Action (Action is basically a function of Kinetic & Potential energy),loosely speaking, least energy in least time.

So, Feynman’s Path Integral approach which also the principle behind Feynman Kac approach for solving Differential Equation originated when Richard Feynman tri s to show that Schrodinger Wave Equation in Quantum Physics can be derived from Feynman’s Path Integral approach.

Feynman’s Path Integral tries to sum all the possibilities of trajectory to find out the resultant path and this came up when he tried to explain wave particle duality in context of the famous  Double Slit Interference experiment..

Though this approach by Feynman needs further improvement and clarifications which I have devised but that’s beyond the scope of this paper. Hence, I would keep myself confirmed to what already is well known rather than proposing my own improvement in physics over that in this paper.

Feynman’s path integrals formula is possibly the most powerful theory in physics to describe the laws of Nature. Starting from the Paul Dirac theory that a body traces the path of least action(roughly speaking least energy and time ), Richard Feynman derived Path Integral approach to discover the path of an object. It basically extends Principle of Least Action to Quantum world from Classical world.

Below are the equations of Feynman’s Path Integral theory to Principle of Least Action which tries to minimize the Energy Function of a system from which are  linked to the Laws of Motions

 

 

 

 

Principle of Least action :

is Kinetic Energy while .

 

Still Science has lot to resolve like Locality-at-distance, Bell’s theorem, EPR Paradox (this year Nobel was awarded on the same Bell’s theorem) for some wonderful experiments.

I could humbly try to take Science to much deeper level to explain my own findings here but that I would deal with separately  the mystery of  Locality-at-distance which Modern Science has not perhaps imagined so far...

 

But what is relevant in Finance context that Market  and hence the dual game play of Uncertainty & Predictability would keep on going endlessly until investors. Market is essentially the sum of Quantum Human Behaviors of Buyers and Sellers in the Market and Stakeholders etc.

Hence, while Market/Finance  has to be studied Scientifically in Causal way, one has to acknowledge that Science won’t do magic  by being able to Predict the Fundamental Uncertainty inherent Everywhere. But there is a good news to deal with Uncertainty which I would cover in different section here on Randomness.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.Randomness (Unpredictability, Uncertainty)Vs Determinism(Predictability)

For Long there has been serious conflict going on in Science/Philosophy/Social/Financial domains whether the world is Random or Deterministic. Science is still far from solving this. But let me state that there exists Fundamental duality in the Universe in terms of Randomness & Determinism just like Light behaves as Wave or Particle , which is fundamental duality inherent in the Universe. Similarly , there exists Fundamental duality of Uncertainty & Determinism in Nature and that’s well reflected everywhere including Financial Markets. Market essentially also has this duality. And this exists Relatively to the Observer. Same thing can be Random or Deterministic to different Observer relatively. Information availability is also one of the causes.

At the same time,I must categorically mention that Causation & Random are not two opposites rather even Randomness has the Cause. But knowing that something is Causal doesn’t mean it’s completely deterministic. It all depends on the level of information. Like we know the Causal Law of Motion of a Car but that doesn’t mean one can completely predict an accident !!

So, Randomness is NOT Randomly Random but “Deterministically” Random. ! Things are Locally Random but Globally Deterministic in Nature, Universe and even Markets. One can relate this to Quantum Physics where a particle behave randomly at individual level but highly deterministic at group level.  

Hence, contrary to the understanding that Randomness is opposite or different from Causality and Causality is overestimated by Humans, I would rather say  Randomness does have Causality and even Randomness has inherent Determinism that need to be discovered but that doesn’t mean they would be completely predictable. It’s like Quantum world as explained earlier, Fundamental duality exists. Things could be random and predictable relatively. As in Quantum world, a wave could behave Random/ Uncertain at some level but Predictable as well at some other level l.

 

So, Randomness has to be studied deeply as the part  of Causality but we have to acknowledge it’s existence and uncertainty inherent in Nature Relatively and Things can’t be completely Deterministic or Completely Random. Randomness doesn’t mean anything can happen, it would always be driven by Causal forces ,but that we may not completely know.

Hence, Randomness is Deterministically Random not Truly Random in Nature hence in Financial Markets as well.

As usually understood , Randomness is not absolute phenomenon, same thing can be random and deterministic from different perspectives and availability of information relatively for the same observer and different observers too.

Randomness does converge to its  Equilibrium State ! This is governed by the Law of Energy and Nature.

So, things – One we need to understand the Causality /Determinism behind the Randomness. Randomness does follow some Deterministic aspects but that doesn’t mean it would be completely predictable. That’s the key to success in real world  It’s linked to attaining more and more Convexity. Will be dealt in the later section.

 

 

 

The Philosophy behind Prediction : Trying  to make the world Deterministic

 

Almost all the financial models try to predict the market based on various statistical and mathematical tools. Even in Life, we want to predict and also in various other social and other domains. Everyone talks about Prediction more or less. But have we ever thought that what we inherently do when we try to predict. So, what does Prediction mean scientifically? Prediction means the the observer is trying to make the world deterministic by knowing the state of the world at Time t   at present (Time =0). Even Nature doesn’t know what it could be at Time T =t because it’s itself a work-in- progress driven by Causal forces. And the State of the world   at Time T=0 and the Time T=T both could be  different and even Nature doesn’t know what could be. So, by trying to predict, the observer might influence the system by influencing itself. These all are Scientific in context of Quantum Reality in Physics.

But is Universe Really Deterministic Completely ? Is it not against the Quantum Law of Nature ? Let’s imagine the world is completely Deterministic or Random ! Will the world run ? No, it will cease to run eventually. The only way the Universe, Markets, Nature,Life could Function is the existence of Duality & Opposites.

So, by Trying to Predict more and more, we are trying to make the system more and more Deterministic which it’s Fundamentally not. Infact there is paradoxical underlying truth .

The more an observer tries to predict, the more uncertainty it would gets. The less he tries to predict, more certainty would the things become. By predicting more and more, the observer tends to change its own predictions using quantum effect. As in Quantum physics, Observer’s role is critical and could influence the outcome. Recall Heisenberg Uncertainty, to Schrodinger’s Cat to Quantum Entanglement in Quantum Physics.

In Other way, to predict like Newton’s Laws of Motion in Physics, one needs to have the information about all the causal forces. But does one have this, No ?

Infact there is a Self-referential issue as in the famous Godel Incompleteness Theorems.

In Science or in Markets, the fundamental issue is that Observer tries to Predict the system,who is itself the part of the system. The System is not independent of him/her. So, scientifically, Predicting the Market/Nature/Universe/Life is Causally Predicting the Self ! Now that’s Contradictory! Can we predict ourselves ? How far ? Market is essentially the summation of Human Behavioral Forces. If an observer can’t completely Predict itself, how come it can predict the Resultant Sum of All the Humans completely?  Caution : I don’t mean own should not predict at all or the world is is not at all predictable. Because, if Everything becomes Unpredictable/ Random, that would also cease the system to run ! The point is observer needs to appreciate and admire the existence of both the sides and act optimally.

 

So, the key is not to try to predict more and more rather act in a balanced way appreciating the principles and Laws of Nature.

This opposite forces or duality creates the potential to run the system. We often see opposite school of thoughts people. slamming and criticising others to prove their own point.

But the highest level of intellectual maturity and consciousness and knowledge is the acceptance of this duality and existence of opposites at the same time. Yes, this Opposite is the source of energy potential that induces the current flow otherwise everything will stop. This duality is inherently the part of the Nature scientifically.

 

The Beauty of Nature/Life/Markets/Universe is Duality and that’s essential to create Potential energy to function

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.Connection between Science, Human Behavior & Financial Markets.

Markets are driven by Human Behavior which essentially follows the scientific  Laws of Nature. Human Brain is run by Neurons, Electric Signals etc. Market is essentially like a Quantum wave. Infact Human  itself behaves like a Quantum wave scientifically.

Science which is often meant Physics has some fundamental difference with Human. Unlike Physical Objects, Human Behavior has some different Behavior but they are indeed linked to each other. Human Brain is also run by Neurons, Electric Signals, Energy etc. Let me state that like Physics, Human Quantum world also has Inertia,mass,weights, momentum, force etc. Broadly speaking just like Classical world, Quantum world also has the equivalent concepts. Indeed, the Newton’s Laws of Motions, Laws of Energy,Principle of Least Action (But Least has different meaning in Human Quantum world) are all valid in Human world. And as a result, even Markets has Momentum, Mass, Energy, Gravity, Acceleration etc.What we call Stocks also have mass, acceleration etc. .  They also follow the Laws of Science/Energy etc.

It has been traditionally told that Humans are different from normal physics. Indeed that’s true but even Humans also follow the Laws of Nature and hence the Markets.

And , for that reason markets have to be studied from that scientific perspective of energy, mass, momentum, acceleration etc. Infact I would explain later how the scientific background of concepts like Large Cap, Small Cap, Mid Cap etc.

 

 

 

 

 

 

 

 

 

 

 

 

 

5. Science behind Black Swan & How to deal with it in Real World ?

As we have seen earlier, the Scientific Universe /Nature has Randomness & Predictability as inherent duality like Wave-Particle Duality. Despite the fact everything is governed by skem Causal forces, there exist many Causal forces governing the dynamics and beyond a observer level of information relatively. So, linked to Godel Incompleteness Results in Mathematics,  In any system there will always exist the zone of Unpredictability as well as Predictability and they exist together. There will practically always be some information not available to an observer at any time.

Black Swan emerges out of this zone  of Uncertainty & Unpredictability which is inherent scientifically in Universe, Nature, Life and  hence in Financial Markets. This would exist in relative sense not Absolute sense depending on the Level of  information available at a time. Hence, Technically Black Swan is nothing but exists in the domain of “Unpredictability, Uncertainty”. Hence , technically the statement “ Predicting the Black Swan is Contradictory as it means Predicting the Unpredictable”. “Predicting the Uncertainty”. I again emphasize it depends on the Level of information available to an observer.

So, what needs to  be done in Real world to Black Swan type events. First of all, one has to understand that it scientifically originates in the Unpredictability/Uncertainty zone.So, Predicting the Black Swan means trying to Predict Unpredictability! Contradictory! If it were Predictable, it won’t have been termed as Black Swan technically.

So, one has to understand the science of Uncertainty/Randomness . As explained in the Randomness section, even Randomness has deterministic hidden pattern and things do converge to their  stable  equilibrium state over the time . This is according to the Laws of Nature/Energy. So, Rather than wasting time to predict Black Swan events, one need to focus on managing risk to be able to survive/ benefit whenever  they come. This can be done by attaining  Convexity ! This is because post Black Swan ,it would converge to the equilibrium state. Those who blow up during Black Swan events are not risk prudent for example  they  are exposed to risk due to leverage or similar fragility etc. The key is to survive during Black Swan. Hence one needs to be Convex enough to survive . This is the part of strategy how we could be more and more Convex.

 

6.Background of Factor Investing

 

Factor Investing has traditionally been nothing but a application of Linear Regression Analysis tools on the Historical Return Data. It’s typically associational rather Causal where Regression is conducted to calculate some superficial not-necessarily reliable relationships like beta and so-called Alpha. Especially Error term is often ignored. Factors such as Value, Momentum, Size etc are quite prominent based on some economic opinions which are not necessarily scientific.

One of the most buzzing factors  has been   Value, which has not performed over the years recently causing a global fuss if really Value matters or it has died.. So, how Value is calculated.

 

 

 

 

 

 

7.Application of Statistical Tools in Finance : Misunderstandings : Role of Time in Mathematics/Statistics in Real World.

Statistics Needs to be Scientific: They are always Backward.

 

Statistics is heavily applied in the world of finance. Almost all the financial models so far uses statistics whether for risk management or prediction etc. But the fundamental issue with all the conventional statistical models is they all are backward looking in the direction of time. This ignorance of understanding the role of Time dimension has possibly  made the Financial models a scam in itself any be. Mathematics has always been studied independently of Time assuming absolute for all. This Platonist view is the cause of misunderstanding  and misapplication of Statistical tools  in Finance which suffers from Hindsight Bias.

Financial Statistical Models are always built in Past data in the direction of Time. At Time T=0, a modeller does curve fitting for Data over T< 0.

But the Modeller doesn’t understand ignorantly or by inertia that T> 0 & T<0 are not the same in the world of Statistics/Finance.

By looking Backward in the direction of Time, Modeller rules  out the Randomness component which is present while looking in future way back then. Future Direction of Time has many possible paths randomly but Backward Direction of time has just One Deterministic path which actually occurred and on which Data fitting is done. This foundational blunder of ignoring the Randomness in Backward Direction of Time is the core of all the issues in Financial Modeling based on Statistical Tools. The entire estimation of Risk, Prediction etc based on such Statistical Tools ignore the vital role of Randomness which a trader actually undergoes while taking decisions in real world. Unfortunately, Out of many possible unobserved paths in future, the statistical modellers only takes the observed path in the backward direction of time.

The fundamental issue with Statistics is that it is  always done in the backward direction of time  based on historical data. That’s where the role of Causality & Science comes in. Causality tries to look forward using scientific mechanism . Let’s imagine in physics, we are using statistical analysis of past trajectory to trace out the future Trajectory ? Does it sound awkward ? So, then how we do that in Finance ? Finance has to be studied like Scientific Physics and that Principle in forward direction of time .  This ingrained psychology of statistical analysis in finance should be replaced by scientific Causal anlaysi of future where we could study Randomenss, Determinism, Human Behavior etc..And yes, some investment and risk strategies are causal based on scientific principles. We will talk about it later.

Hence Psychologically engrained  Backward Time Looking Statistical Analysis must be discarded and Forward Looking Scientific Causal Analysis like we do in Physics etc must be adopted. This would also be the key to Scientific Risk Taking & Management and dealing Scientifically with Black Swan type events in Real World.

 

The tools like Backtesting etc are the subset of that unscientific understanding of Role of Arrow of  Time in Statistics in Real World.

 

That’s the reason why Finance has to be developed in forward direction of time as we do in Physics. Do we Backtest in Physics to predict the Trajectory of a vehicle /Car or we study the equations of of causa forces. This fundamental psychological change has to be brought in the world of finance where the trajectory of a stock price is studied scientifically by analysing Causal forces, rather Backtesting.

Even for those who simulate Randomly in future like MC Simulations etc must understand that There could always be more Scenarios in Real World than one can simulate using Computers or otherwise. This is fundamentally related to Godel Incompleteness Theorems.

 

The point is Finance has to be studied in scientific way in forward direction of time like we study physics. That’s the way, we can have better understanding of risk-return in real world finance..For that ,we will have to understand the Causal concept of Uncertainty/ Randomness in Life/Nature/Markets. This is is because by understanding this concept, one can take Scientific decisions to build the portfolio or otherwise.. Otherwise all those Statistical based tools like Sharpe Ratio, Drawdown, Correlation etc. Out of historical data or observed data is just the tiny subset of all the possibilities and hence hugely misleading for understanding risk.

Like in Feynman Path Integral approach of Quantum Mechanics /Paul Dirac Principle of Least Action, Explore all the paths not just the observed one.

 

 

 

 

8.Regression Analysis, & Paradoxes

                                   Y = 

       

 

Let’s assume the Linear Regression Equation

                                          

 

 

       

 

Where Y is the Dependent Variable , X is the Independent variable and α is the Error Term of the Linear Regression.

                            X =  +*Y +

 

So, we can see that Association ignores the Direction of Time here. . It could be both way But Causation has direction like Cause and Effect at different Times subsequently.. ! To determine Causality, we will have to block all other paths/factors and test the direct effect ..for example…Stocks in different economic scenarios…or other company specific factors to test the causality …

For example: The Sun rises in the morning and the bird sings in the morning .They are obviously correlate but to test if Bird singing causes the Sun to Rise  We can test if the bird sings when the Sun doesn’t rise on cloudy days or if the Bird doesn’t sing, does the Sun rise or not… We would find that this is not the case. That  means it’s association ! Like two cars running on road would be associational not causal and they could suddenly change the direction after sometime if roads diverge suddenly…So, misunderstanding those two cars relationship is causal could mislead suddenly! Similar is like Factor Investing. If  certain factors work doesn’t mean they are causal..they could be misleading and could be by chance unless Causality is established !

If this is causality the absence of one would definitely affect the occurrence of the other event. If A causes B then if A doesn’t occur, B should be affected every time. That need to be tested.

But we do have to accept that there could be many more unknown causal factors that we might not know which could be in the form of randomness errors…!! So, that’s why RCT type approach would be required at later stage to deal with them !!

 

 

 

Point 1) In the first one variable Linear Regression Model (LRM), what is the most important term is the Error term(residual ). This represents the Randomness ,Uncertainty component . The error term is the TRUE Origin of the Black Swan events !

 

What is to be noted that in the above Regression formula, X1,X2 etc. are independent variables usually ignored in Factor based Investment  Models.

For example, while doing Regression analysis in Factors, why is it assumed that different factors like Value (HML), Momentum,Size (SMB) etc are independent ? Are they really independent ? No , so how far this Regression Analysis application to calculate Alpha is accurate ?

 

1)Reverse regression Y on X and X on Y are different

2)Relativity of Beta when many independent variables X 1 ,X2 etc are included. Beta changes …

 

Statistics  is NOT wrong but Statistician needs to understand the statistical terms well in real world applications.  This error term which is often ignored is the most vital component in the real-world financial applications. This is where the Mystery lies !!

 

 

 

Structure of Machine Learning Models Like the Regression and other ones :

 

 

It’s all about finding the suitable functions . Here also once can see that though could be many but the error term still exists which is the source of Black Swan Tail Risk.

 

The key issue is while most approaches on finding f() based on historical data fitting, what should be the focus is on managing future error terms (randomness uncertain components).

That’s where the real world model development is needed…Managing the Unknowns and Random Components…

 

Can you predict your own life using Regression Analysis ? How far ? Market is essentially the resultant of all Human Behaviors ! Need not forget.

As Nassim Taleb says Real World is more Random than RealWorld. Indeed that’s right !

One more fundamentally important point is that Stock Prices don’t move on Euclidean Space of Paper in reality . This is virtual trajectory. In Real World, the Stock Price moves in different Non-Euclidean /Riemannnian Economic Space-Time which captures Human Quantum Minds etc.. Then we are trying to superficially/virtually draw the trajectory on the two dimensional paper . So, while doing Linear Regression Analysis , it should be independent of the intrinsic characteristics of the underlying space. Hence, whether Regression Analysis is done on Euclidean  or different Geometric Space (as in Machine Learning etc.) the true physical relationship must be invariant. Hence, these parameters like Beta etc depending on the slope of line in the Euclidean Space-Time could be fundamentally misleading !! Because in different Space-Time, Beta (calculating using Euclidean Metric Least Square Distance )could be possibly different but it has to be independent of the underlying Space. This is extremely fundamental issue while doing conventional Regression analysis . In a nutshell, this Beta relationship couldn’t be in the Real Economic Space Time but the characteristics of the space of the paper on which we have superficially assumed to draw the trajectory of the stocks !

 

REGRESSION PART2

Regression Part 2:

 

A)Causation is in the dimension of time which LS Linear Regression doesn’t take into account….Time dimension…it treats them statically in timeless dimension only space

We have to understand in real-world perspective that Causation is established in the Space-Time dimension not just Space. As in the Physical world,  cause occurs at say for example T=0 and effect occurs at sometime in future T= t. If we ignore the dimension of Time and take only Spatial locations into consideration, it becomes just association not causation. For Causation we have to test the Cause and Effect in Space-Time not just Space. Unfortunately, LS Regression is being conducted particularly in Factor Based Investing and otherwise in Finance ignoring the Time Dimension. This omission of time makes the entire relationship like associational of patterns that could be just co-incidence but not scientifically causal.

By running the following LS Regression

 

We often inherently assume that X  and Y both  are able to transmit information causally “simultaneously” at time T=t  at more than the speed of light, but how this is possible ? If they are really causal, then first X should occur at Time T=t and this should cause effect to Y at some time in future T = t+n where n >0.  But  in the conventional LS regression it is inherently assumed that X and Y both are cause and event at the same time which is contradiction in real world scientifically in Nature.

It otherwise proves that X and Y relationship is Associational based on Superficial Patterns either caused by Coincidence(Like two independent Cars moving on the road in the same direction misleading an external observer to be causal to each other) or they are caused by some hidden causal mechanism of some other variable known as Confounder. The above LS based regression can take place at the same time “Simultaneously” iff there is hidden Confounder(if it’s not coincidence!) as information can’t travel at more than the Speed of the Light ! (Of course it’s not Locality-at-distance logically!)

 

So, for Causational Proof, the Equation has to be in the form of do calculus. This is causal intervention  that if X is causally set  to value x at time T=0 for example how the effect on  Y behaves in  future value of Time =t  and that time t can have different values on case to case basis.

I have been advocating for long that Time Dimension is often ignored in the Statistical and Mathematical world(Timeless Platonic world) unlike Physical world(Space-Time). The role of time appears non-sensical often but at deeper level it makes huge difference! Many of the Paradoxes in Mathematics originate also due to this omission of Time as well fundamentally. Say for example, Theory of Relativity considers “Simultaneity” as Relative while Conventional Set Theory in Mathematics rests on the Principle of Simultaneity as an Absolute Phenomenon. This is beyond the scope of this paper. But I mentioned to show how this foundational misunderstanding percolates down to statistics and Association in LS based Regression is often misunderstood as Causation !

B) Let’s go back to the history if LS Regression method. It was first formulated by Mathematicians like Guass, Legendre in 1722 to around 1800 for the estimation of  Trajectories of the Celestial Physical Bodies like our Earth in the Euclidean Space.

That’s  mathematically good estimation as that physical space-time behaves like Euclidean Space-Time. But the foundational issue arises when that Tool from Physical Space Bodies is being applied to the Financial & Economic Space-Time. The financial and economic space-time driven by quantum human minds is not essentially  an Euclidean Space-Time rather they have Non-Euclidean/Riemannian or some other Space-time in Real World. Statisticians import the observations from that Space to Euclidean Space of Computer Screen or Paper to draw and apply LS Regression ! But this raises fundamental question that is the Euclidean Space of Paper and Metric calculation doesn’t bias the Estimation of Financial & Economic Variables from a different Non-Euclidean space-time ? I am stating this because the way Least Square method is developed, is it Not inherently  dependent on the Euclidean metric relationship ? What if these data are drawn on some other Non-Euclidean Space ? I mean this Relationship should be independent of the Underlying Geometry of Space where these are drawn. The Real -World Economic & Financial Space-Time is not Euclidean !! This a point of exploration as even in ML methods, often Euclidean metric tools are applied. But it could be Non-Euclidean and other Riemannian Metric Spaces as well to better discover the relationship.

 

But anyway let’s confine to the Euclidean one that is generally taken to derive the Least Square based Regression Method for a while ….

Least Square based Regression Tool is due to the algebraic structure of Least Square Formula where it minimizes the Sum of Squared Differences(we can say them Euclidean Errors!). If we change this objective of Minimizing the Squares to something different, the entire Calculation of Beta would be different and the value of Beta would change ! Beta is actually dependent on that Euclidean method. But why we minimize the Sum of Errors is itself under question ! How far is this process effective to take into account the Outlier in Real World ? Let’s say we don’t minimize the Sum rather we minimize some other functions that could be more suited to Outliers for example from Tail Risk Perspective. There could be different values of Beta on the same set of Data the way we define the minimization function of errors. It’s not Absolute ! It’s Relative ! The important concern is that in Factors Investing and Finance & Risk Management LS Beta is blindly applied for Allocation, Risk estimation that has proven to be disastrous in Real World  in case of extraordinary(outlier) scenarios.

 

Further, one more important aspect of LS based Regression is the Exogenous Condition that

   E[error] = 0

And if this exogenous condition of error is not satisfied, the whole estimation of Beta is unreliable and biased ! This is the fundamental reason why in Real World Finance, Beta is often biased and variable in Financial Time Series data.

Exogenous condition means the independence of Error terms E[error/X). If Exogenous conditions are not satisfied, that means the Error itself is the function of some hidden relationship . It could be that Error is  dependent on Y or X or some other hidden variables which govern the dynamics of the error and which in turn make Beta unbiased and unreliable as both are mathematically related ! hence the internal dynamics of Error has to be established otherwise it would affect the other aspects of the LS Regression. Even there can be different dynamics of Error say if Error itself is some other variables regression

If such is the case value would be highly misestimated ! This actually happens in the Real World Finance & Economics Time Series data !

 

It’s like in algebraic equation

In this case the  would be highly misleading ,biased and incorrect mathematically.!

Infact such conceptual mistakes highly mislead in the real world finance and economics where beta(technically slope) becomes too variable dependent on the data time frame etc…and exposed to Black Swan Tail Risk affecting Trillions of Dollars of Investment globally affecting common people and Investors’ lives.

Error basically means Random Component ..all the deterministic components of Y dependent on Independent variables like X  have to be removed from the Error term.

So, In Financial /Economic world, unless Error satisfies the Exogenous condition , the LS estimate value of will not be unbiased.

If the expected value of the Error term above is not 0(the violation of Exogenous condition ), it means that there is still some hidden deterministic variables relationship to be discovered inside error terms. The error term like constant  as in the equation is not a constant but a function of some hidden variables or even .

The correct meaning of is that uncertain Portion of . In Real World example the error term represents that Unpredictable component when we try to explain Y in terms of X causally.

It's like we intervene by the cause   at say Time T= 0 and measure its effect on at Time T= t. Infact in true causal sense these two occur at different points in time not at the same time(simultaneously) because information traveling takes some time. Then we try to understand what uncertain random component of  Y is not explained by the deterministic components of the equation. We call this as the error causally. But this omission of Time dimension makes the entire thing superficial associational  relationship and just an artifact of Euclidean metric space !! The way error is defined as Y – E[Y/X], it shows that the error term is not independently defined.

 

This is like B1 Spurious case as explained by Prof. Marcos Prado in his interesting paper.

“This is most likely valid in the field for which LS Regression was formulated in Celestial calculation but not  often in the Financial and Economic Space.  The Real World Data hardly show the Exogenous condition Satisfied.”

Econometricians can’t define error deriving from algebraic equation like Y- beta*X rather error terms should independently satisfy exogenous conditions E[error ] = 0

Error should be independent of X

Exogenous conditions means Regression assumes deterministic set up.

Expectations of error or randomness to be 0

Correct meaning is if we do[X] at time t= 0 then at  time t >0,   Y should equal to X*beta.

Error means Unexplained part in Causal relationship of Y due to X

Most Machine Learning Tools focuses on the deterministic part but there is need to focus on the Randomness(error) part and its underlying mechanism and how to manage those unexplained, uncertain components.

Present regression at same time means it assumes the role of hidden confounder.

What is important to observe that in LS regression, the order of variable is also very important.

The two regression lines below are not derived from each other with respective coefficients.

 

That means in general LS regression method

                              

                                   

This reveals the internal dynamics of LS Regression that the Minimizing the Sum of Squared Errors method is not symmetric in this sense. The Order matters and at the same time the values of the Slope and the Error terms depend on the error and are not inter-linked. This implies that in real world financial and economic applications, before applying this associational LS regression, one has to make sure about the Order otherwise the derived slopes and randomness error would change leading to different estimations of risk affecting allocations etc. This also shows that Error itself is relative and order of variable dependent on LS based system . But in the Real World, the Random Components should be linked to each other if the Variables are the same just order changes.

Coming back to the Commentary : the entire system is getting misled superficially by the Associational relationship based on LS type Regression affecting Trillions of Dollars Globally.

The way sum of squared error minimization has been defined in LS system, it’s dependent on the Euclidean type system.

Econometricians assume they are doing Causal but the mathematical tools used are associational.

If LS system is changed, Beta will also be changed..

Question why LS system ?

Rather than minimizing the sum of the squares, there could be other better approaches as well from tail risk perspectives. (This is the other detailed research topic in itself !)

There is no fixed Beta . The value of Beta  would depend upon how the  Error terms are dealt with like LS equation. This is because Beta is derived by minimizing the Sum of Squared Error Terms.

 

Lets say there are different points

 We are trying to find the regression equation

                          where

Here the values of are those which minimizes the function

 It shows here that at time T=0  the error(randomness) terms are figured out, then at time T= t  >0 they are minimized mathematically to derive beta (deterministic)term ! This is also like In-sample derivation of Beta term(Deterministic Term) from the mathematical minimization of  function of Random Terms!

Please imagine the role of Time dimension here

 

 

Lets say the Regression equation

                         

                                      

Here it shows that Randomness Term is the Function of Deterministic Term and that’s why we take the derivatives to find out the minimum.

That mean at Time  T =0, we try to estimate the Error first and then calculate Suitable Beta at Time T= t >0 , Please note the role of Time Direction as well. This is because for Causation, role of Time Dimension is the fundamental requirement in Real World. By the way,  This is like In-Sample Estimation where Beta is searched by deriving from the  Error terms.

Ideally what happens in the Real World… We should pre-estimate Beta(Deterministic Expected Term) at T=0 and then measure Error Terms “INDEPENDENTLY” at T = t > 0 and study the Exogenous Condition E[ error/X] =0 if that is satisfied or not ! In the existing previous approach, “INDEPENDENCE” is completely compromised as Beta is derived from the Error itself !!

This is in principle foundationally incorrect leading to Biased estimation of  Beta in In-Sample way.

So, the entire traditional Beta Estimation in LS approach is like In-Sample. It doesn’t tell about Out-Sample Beta in Real World in Forward Direction of Time! Especially for the Financial Data which is often unstable, this could lead to huge misestimation in real world out sample result.

As we have seen earlier, the LS approach  was formulated for Celestial Physical Bodies which is Classical World Stable Data unlike Financial Data driven by Quantum Human(Trader & Investor) Minds.

 

EXOGENOUS

 

 

 

 

 

 

19. Exogenous Condition :

LS Sum of the error minimum when Expected (error)= 0 . LS approach relies on the Expected (Average ) figure where Average is hugely misleading Statistical Tool in Financial world with fat-tailed data.

The issue is even Exogenous Condition is not sufficient in Real World Finance because even if Expected i.e. Average is 0 satisfied but there is large negative movement and then subsequently large positive movement but the system could get exposed to Tail Risk Bankruptcy in the large Negative Error term and can’t even wait for the next Positive Upside Error Movement. So, from that Perspective of Tail Risk, even Exogenous Condition is not sufficient. It’s another detailed discussion of its own how these conditions should be re-framed !

 

 

 

 

 

 

 

20. Beta is not Absolute but Relative.

Beta depends upon how the error is dealt with and also with respect to what other variables are .

The Exogeneous condition of error is extremely important.

So, probably one test is Take Beta to that level until Exogenous condition on Error Randomness is satisfied independently !!Then only Beta would be unbiased. Otherwise, it means that the Beta is biased as Error Term itself is some function rather independent ! As long as the error doesn’t satisfy exogenous conditions independently, it shows there are some more factors(contributing to different betas … deterministic components) to be discovered….

Moreover Beta value depends on how Error Term formula is tweaked like in LS form, There can be other forms as well. Beta would be different in various cases. It’s all dependent on the method followed.

Beta should be such that it should maintain a balance between tail outliers and normal depending upon the requirements.

If Error has intrinsic hidden further pattern ..it would make the existing beta biased and different types of structure within error would affect the dynamics of  beta and further may be causality or not.

21. Hidden Dynamics of Error Terms :

Now Further, There are different  possible scenarios….Error terms have some hidden Deterministic components which affects X or Y or X affects the error term or even Y affects the error term …in all different cases the validity of Beta would be different

 

Say for example

 

 But if Error term is  biased and Not Random say for example

And further

 

.

In that case, the entire estimation of would be highly biased as this is derived from which itself is the function of different variables implicitly rather than independently.

 

Biasness of Beta

It’s like three variables  algebraic equation in x and y and z..you are assuming z as constant and make quadratic in x and y…it could be technically wrong !! Similarly until error term is exogenous and independent…..it would lead to wrong beta estimation !! That’s why in Regression.. independence is of huge importance.

If Error is function of some hidden variables then Beta won’t be correct…how do  you make error is independent ?

Simply doing regression without independence of error is technically wrong  and could lead to biasness and misestimation

Like in Quadratic or Functional algebraic equations if you treat a variable as constant ..it would give  wrong solution

Y = m X + c ….here c needs to be strictly constant ….can’t be a variable related to X and Y or else m would be incorrect…

One can’t treat function as a constant and do functional algebra ..it is technically wrong

So when doing such statistical regressions, make sure error terms is not a function of some related variables or else beta would be biased and unstable over the time…and as beta would depend on error

That’s the reason Beta  becomes unstable. It can be tested in real world  where we see that the measured value of beta keeps on changing rather than fixed constant ! One can experimentally verify this on many financial time series data and check how beta has its own inherent dynamics. Beta keeps on changing dynamically over the time dimension  or otherwise relatively if new variables are included!

Error would change over the time if it’s function ..not exogenous conditions satisfied…

If Error is itself a function not satisfying exogeneous conditions ..it could be the source of Black swan

 

 

 

 

 

 

 

 

 

 

Error Terms & Simpson’s Paradox : Simpson paradox reveals a lot about the Reality !

 

A) It reveals that the DETERMINISTIC parameters like Alpha, Beta themselves have Randomness over the time !

B) Further, the existence of these parameters and error terms are RELATIVE  !! THE LINE OF BEST-FIT itself behaves geometrically RANDOM over the time ! What was the Direction of Error Terms eventually becomes the LINE OF Best Fit and what was Line of Best-Fit becomes the Error ! This reveals the Fundamental aspect of Relativity and Randomness over the time for an observer.

It reveals the Fundamental Duality of Randomness & Determinism existing in the Universe and hence Markets.

The Universe and hence its subset ,the Markets have both the components: Fundamental & Deterministic. One can’t differentiate between the two. They exist Relatively and Simultaneously. Deterministic(Line of Best Fit) becomes Random(Error term) and Randomness(Error terms) becomes Deterministic ( Line of Best Fit)

One has to be very careful while making decisions based on the Regression analysis particularly ignoring the Error terms ! Blindly following the Fixed  Line of Best Fit to make investment decisions could be highly misleading and disastrous in real world !!

So, this points towards extremely fundamental points while doing the Linear Regression analysis : That parameters like Beta, Alpha etc. could  keep on changing depending on the data set and there are no fixed values.

First, Two things tracing the same trajectory  might not be correlated in true sense. Say for example two cars moving on the road would appear to be correlated to an external observer from sky. But that’s not so, may be later the road diverges after a long time and both the Cars could change the directions and befool the external observer from the sky who was assuming them to be correlated by observing their past trajectory for a long time !!. Similarly in markets !!

The Point is to establish the Scientific Causation first and then one can rely scientifically Regression to some extent as long as error is managed well.

For that causation, one has to block non-causal paths by observing the Y  & X while blocking other variables or checking how Y behave  if X is there and NOT there. If X causes Y then if X stops, Y must be affected or stopped !

###

 

We have to understand that Regression is fine  it could help in prediction at times but given the large unexpected error terms at times could be highly misleading at times and disastrous. So, what one should do optimally is to expect along the Regression Line of Best Fit but still be prepared to manage Random Error term which could be huge in the world full of Complexity

 

 

 

 

 

 

 

 

 

 

 

 

 

24.Equity Risk Premium Puzzle  : It’s  indeed the bug in Factor Investing statistical structure based on the Linear  Regression Analysis which we have discussed earlier…Capturing Human Behavior dynamics based on Linear  Regression analysis deterministically while ignoring the randomness component is deeply flawed !  This puzzle arises because we misunderstand the concept and foundation of Regression Analysis.. Any regression  based model to determine this risk-premia will have fundamental concern… This is simply related technically to the fundamental issues of randomness  and structure with such Regressions on which CAPM and Factor based Models are based… the foundational issue is in the statistical tools itself on which such model puzzles have been developed !!

So, the ultimate solution to one of the most important historical puzzles in finance is to look into the flawed conceptual  misunderstanding of the statistical foundation which has given birth to the so-called puzzle. The origin of the Puzzle is the loopholes and misunderstanding in the statistical infrastructure which has given birth to such superficial puzzle. Once that is understood, the puzzle would cease to exist automatically ! So, this puzzle infact hints to revise the foundation !

I must again categorically mention : When we get the extraneous solution of any mathematical equations, it’s not that mathematics is incorrect rather the assumptions while applying the mathematics are incorrect. Similarly, here we need to revisit the foundation of the flawed assumptions of factor investing and the statistical infrastructure like Regression. Coming To Factor Investing Analysis : As the Byproduct of Regression Analysis

 

Equity Risk Premium Puzzle :

 

The main  reason behind this Equity Premium puzzle is fundamental structure of the Regression based CAPM model, Misestimation of Risk Calculation and Ignoring the Time Dimension.

Often Risk is measured in terms of Beta based on Historical data in backward direction of time using the same flawed estimation of LS regression techniques. Even Traditional Tools like Max Drawdown etc doesn’t capture the true Risk in the Real World. The fundamental reason is Time dimension is ignored. If we look in forward time direction, there could be many possible scenarios that an investor is willing to take while investing in Equity but not all such possibilities actually occur. Say for example an investor in the time of Uncertainty is willing to take risk of even market falling -50% say due to some war or some viral outbreak etc.. But somehow, market didn’t fall that much and only -25%. So, when regression  is done on historical data or even max drawdown is calculated on historical data in backward direction of time,  this possible forward scenario of -50%(which didn’t occur) was not included ,which an investor actually took in forward time uncertainty.

Existing Conventional Tools based on Regression Analysis and Max Drawdown, Sharpe Ratio etc. often don’t  take into account the possible future real risk which didn’t occur but was a possibility in forward direction of time ! infact that was the real-time risk an investor took in forward time direction , but was omitted in backward analysis of time ! This could hugely underestimate and overestimate the risks.

So, the fundamental reason of Equity Risk Premium is the

·      Misestimation of Risk in Real World by Ignoring Time Dimension and Traditional Backward Looking Tools e.g. Beta, Share Ratio, Max Drawdown etc.

·      Possible Randomness in Future gets omission in Backward Analysis!

·      Fundamental Issues with LS -Regression based approach (the backbone of Traditional Factor Investing)

·      Also, the omission of Fat-tailed Outlier data which could drastically change the  estimated risk parameters like Beta etc..

Hence, unless the Risk is Truly measured in Forward Direction of Time( Not on Historical Tools), the true estimation of Equity Risk Premium would be inaccurate.

You are measuring the estimate based on back ward looking data….in forward real world risk is way more than backward looking due to Uncertainty…

 

*Because equity risk premiums require the use of historical returns, they aren’t an exact science and, therefore, aren’t completely accurate.*

 

The estimates also vary largely on the time frame and methods of calculation…removal of outliers fat tailed negative data

Relying on CAPM type regression model to do  the historical analysis for equity  risk premium based on Beta etc.not reliable enough

Risk should be truly seen in the forward direction of time not backward analysis of Historical data using regression models

In real world we see how much risk we take for equity investing over the Risk-Free rate etc.

Measurement of Risk in the form of Beta using Regression on historical data is not correct in RealWorld!!

Real world Risk in forward direction of time is way more than the backward analysis of risk using regression tools

Lot of randomness that appear in forward direction of time in real-world doesn’t get captured in the backward analysis

For example : risky scenarios  that could have taken place while taking investment decisions  in fhe forward direction of time ..but say they didn’t actually happen …will not get captured and lead to less risk calculation on historical data  regression analysis…

.: Misestimation of  actual risk in forward direction

·      Risk is hugely misunderstood on backward Anlaysis and in forward directions..many disastrous possibilities had the chance which actually didn’t occur and hence don’t get captured truly in the risk measurement.

An equity risk premium is based on the idea of the risk-reward tradeoff. It is a forward-looking figure and, as such, the premium is theoretical. But there’s no real way to tell just how much an investor will make since no one can actually say how well equities or the equity market will perform in the future. Instead, an equity risk premium is an estimation as a backward-looking metric. It observes the stock market and government bond performance over a defined period of time and uses that historical performance to the potential for future returns. The estimates vary wildly depending on the time frame and method of calculation.

Future Beta & Historical beta

An equity risk premium is an excess return earned by an investor when they invest in the stock market over a risk-free rate.

This return compensates investors for taking on the higher risk of equity investing.

Determining an equity risk premium is theoretical because there’s no way to tell how well equities or the equity market will perform in the future.

Calculating an equity risk premium requires using historical rates of return.

CAPM model itself needs fundamental change based on historical regression

How exactly to calculate this premium is disputed

How exactly to calculate this premium is disputed

Equity Risk Premium = Ra – Rf = βa (Rm – Rf)

But how do you calculate future beta in real world   ? Not based on historical analysis

By the way this is associational not causational …

Expected equity risk premium likely depending on the hidden causal factors not just on this associational beta based on Regression where the role of confounders also come in.

Future return Equity Risk Premium would actually depend on the Causal Mechanism & factors  & Randomness not this associational type models like Factor based models and CAPM etc.

Future Expected Fair  Valuation drive the return causally scientifically Not this CAPM type associational superficial statistical theory

One needs to further scientifically  understand why an investor took systematic risk means he/she should be compensated with return why ? What is the science behind this  ? This would be explored later.

Missing Causal Confounder variable actually affects the equity risk premium. That is hidden but could make the entire risk calculation biased !

 

 

9.Newton’s Laws of Motion in Markets & Quantum Law for Markets

 

Newton’s Laws of Motion for Markets : Principle of Least Action by Paul Dirac : Feynman’s Path Intergal approach in Quantum & Classical World

Laws of Motion of Stock Prices in Quantum World:

 

Net Resultant Valuation(Energy) generates Forces(Demand * Supply i.e. Buy and Sell Orders i.e. OrderFlow which finally leads to the Motion i.e. Change in Momentum

 

 

 

 

Newton’s Laws of Motion in Microscopic world :

Consider a system of N classical particles. The particles a confined to a particular region of space by a "container'' of volume V. The particles have a finite kinetic energy and are therefore in constant motion, driven by the forces they exert on each other (and any external forces which may be present). At a given instant in time t, the Cartesian positions of the particles are r1(t),⋯,rN(t). The time evolution of the positions of the particles is then given by Newton's second law of motion:

mir¨i=Fi(r1,⋯,rN)

where F1,⋯,FN are the forces on each of the N particles due to all the other particles in the system. The notation r¨i= d^2 (r i/dt^2) i.e. second derivative.

N Newton's equations of motion constitute a set of 3N3 coupled second order differential equations. In order to solve these, it is necessary to specify a set of appropriate initial conditions on the coordinates and their first time derivatives, {r1(0),⋯,rN(0),r˙1(0),⋯,r˙N(0)} Then, the solution of Newton's equations gives the complete set of coordinates and velocities for all time t.

For an Investor , based on the valuation, the causal force of demand is created for a stock by the trader/investor. The Valuation is like Total Potential (in Physics) that generates the force in the stock.

Valuation is relative depending upon the investors’ perspectives. Valuation(Potential) generates the market forces. Yes it’s as per the  Physical Laws of the Nature.

So, there would be different valuations for different investors/traders (who are like human brain quantum particles!) causing different forces.

 Let  denotes that equivalent force caused by the respective potential (Valuation)

 

 

So, the causal process : market is the system of different investors

Each investor has the relative valuation . Every Non-Zero Valuation would apply  either upward or downward forces. 

 

Causal Process due to Forces in the Markets.

Different Valuation-generated forces   attracts  and causes different investors/traders to put Buy/Sell Orders  .

(i.e. demand and supply forces) .This is scientifically  like the gravitational force on energy or law of attraction! Higher Value tends to attracts small values towards itself..

The Resultant Order Flow Imbalance is caused by the Sum of all these Buying and Selling Orders(Caused due to the Value generatedDemand-Supply Forces).

is like Electric Flux i.e. Number of Order Lines passing through due to Valuation Caused Field.

(Note : I refer here only to the genuine orders intended to be executed not the false orders ,for that there will be more causal analysis)

 

 

The Resultant Order Flow( due to Valuation Generated Forces) causes the Momentum change for the next time duration [Newton’s Laws of Motion]                              

                                    

 

 

 

 

 

Now As the Stock Price gains Momentum due to  forces, Price changes using F = m *a (Newton’s Laws of Motion type mechanism), The Price changes subsequently.

Price at time t+ would be caused by Momentum generated at time t  and then updated Valuation for different investors would change relatively due to change in Price and Momentum in the dimension of time. Price and Momentum being two of many factors affecting the Real- World Valuation at anytime. ( This has been explained in detail)

 

                                

 

 

Explanation :

Valuation at any time is the function of many multiples and factors and their dynamics over the time  for every investor. There is a resultant of all the Valuation. Open-ended process.

 We have explained earlier

 

So, to be specific related to Momentum and Value Relationship

a) Valuation at time t is also dependent on Momentum at time t-1 i.e. previous momentum. The mechanism for Valuation is mentioned below..

a)Valuation at time t drives the momentum (velocity) at time t for t+1 through the force it generates.

                                   

a)As the Stock price goes into the motion(momentum), it adds values for different investors leading to change in the Order Flow Imbalance.   is caused by more and more forces in the markets caused by the Valuation changing  over the time.

Number of Field Lines crossing is akin to Number of Trading Orders of Traders/Investors caused due to Valuation changes as the Potential.

Which depends upon the Current Strength caused by the Potential/Force/Valuation changing over the time.

Order Flow is essentially and scientifically  the Market Quantum Force. One needs to study the Order Flow (Force) to be able to understand the motion of the resultant Stock Price. Like we study Newton’s Laws of Motion to figure out the trajectory of a body. One can apply ML to study Order Flow Dynamics to study the Quantum Forces driving the Market.

So, Based on the Laws of Quantum Motion affecting The Human Traders’ Behavior and hence the Stock Price can be studied in the ORDER FLOW dynamics how Forces evolve. This is where ML could be useful. I state and claim based on my experiments that it’s similar to Energy system of Particles. This is linked with  Century Old Problem and essentially  Financial Brownian Motion would be similar to Geometrical Brownian motion but in different sense ! This is linked to what I’ve explained in other parts that Human Brain follows the Laws of  Quantum Motion and that’s the advanced version of Classical World Laws of Motion. So,microscopic Newtonian dynamics, the Boltzmann and Langevin equations are derived for the mesoscopic and macroscopic dynamics,

Hence I state and claim that one can experimentally verify that Financial Brownian Motion (Quantum Living  version) is  more advanced form of Physical  Brownian Motion (Quantum Non-Living Version) although their foundations being the Energy system only  fundamentally but in different forms. This fundamentally originates from the fact that Human Brain Quantum Behavior is the advanced version of Classical Laws of Motion where the former is variable while the later is constant. But yes all these Classical Laws of Motions are valid and originate from the Quantum world only. That too, Living Beings Quantum Laws have more variable components than Non-Living Quantum Laws but all of them are highly Causal. This is beyond the scope of this paper.

But for this paper, that can be well studied and verified in the behavior of Order Flow dynamics  ! It’s fundamentally the energy dynamics.

More the Valuations for different investors, it would attract more OI assuming there is availability of fund with the investors.

 

To note : Momentum is just one of the factors among many. What matters is the resultant of all like in physical forces. These are similar physical  forces attracting investors through their quantum force on  their brains(neurons).

 

Step2  :At any moment in time The V(upward/buyorders )tries to push the price up and V(downward/sell orders) tries to push the price downward.

This is tussle between the two driving the market ,eventually converging to the equilibrium state.

Step 3. The Momentum drives the prices further up due to the net upward force. Gradually based on the combined effects, the price moves.  The distance travelled by the stock price can be calculated on the change in momentum caused by the forces.

Step 4: As the price moves up and the momentum increases, it affects the Valuation dynamically. Valuation (upward ) and Valuation(downward) keeps on changing time to time.Upward force and downward force keeps on changing. .Gradually as the price goes up significantly , the Upward Force weakens gradually due to weaker upward Valuation and downward force becomes strong due to stronger downward valuation. Upward That leads to price coming down again. This cycle goes on to  reach the state of equilibrium and fair value over the time.

 

 

                        

[Newton’s Laws of Motion] This is universal law in Nature not just for Physics of Non-Living Objects !  But yes in different forms because nature of motions differ in classical and Quantum worlds ! But this is also applicable in quantum brain world ..

 

 

 

 

 

Here Mass of Stock can be taken as Market Capitalization which typically means Small Cap Stock will have less mass than Mid Cap than Large Cap. So, why Small caps could be more volatile than large cap in general is also because of their lower mass and hence requirements of forces. But this would depend on the resultant forces on ase to case basis. But yes it’s indeed caused by  scientific equations  of forces !

Like in Classical world, Quantum world also has gravity ! Larger stocks are having more mass and hence gravity than smaller caps in general comparatively. Yes it's like Physics . So, there is more amount of force required to move its price to similar distance (i.e. same return ) compared to small cap (low mass) stocks in general given all other  things are constant. Hence they also show lower risk as more amount of force  would be required to pull their prices down. And there this forces come from demand supply forces which are actually quantum forces in the brains of buyers and sellers ( traders) or computer codes…

 

“Factors have to be discovered scientifically how they contribute to the Quantum Laws of Motion of Stock Prices  not associational regression  based approaches traditionally followed.   One has to discover how a factor is affecting the acceleration/motion of the stock through Force and Mass ..For that one has to study the Law of Quantum Motion of a Stock Price originating from Human (Investors) Forces of Demand Supply & Order Flow. So, one has to study how Momentum , Size, Quality,   etc. are contributing to the Scientific Laws of Equation .

Not that they should be compensated for their risk due to a factor and hence the return.

This Economic Rationale that an investor should be compensated just because he/she has taken some specific type of risks has to be scientifically explained and determined how they are contributing to the Laws of Motion of a Stock Price, otherwise it’s just a superficial and non-scientific. Infact explained later that Economic /Financial Space- Time itself exists scientifically having their Laws of Motion as we have in Physics!!

 

 

 

The Issue with Conventional Value Method

A commentary on Prado’s calculation of Goldman Sachs Factor Index Calculation of Performance

Connection with Regression Analysis Price to Book Value wrt.  ROE (Aswath Damodaran).

This Regression Analysis is again applied with misleading results about Undervalued and Overvalued. Also, something becomes undervalued using one multiple while overvalued using another multiple. It also reveals the flaw and incompleteness of single multiple approach to Valuation. If one uses(despite flaws),one should possibly  take into account different multiples to figure out the same.

 

Traditionally we have seen factors e.g. Value, Size, Quality, Momentum etc. based investing being very popular and at the same it has also shown the dismal performance over the last few years.

Two important aspects :

a)Value as the term has been defined incorrectly in this context that Low price to Book Value mean Undervalued and High mean Overvalued or similar other analysis. Infact this decision on Low or High is based on the LS -based Regression Line  as well .

Given the fundamental issue with the Regression Line, the basic definition of Low and High could be mis-estimated.

Moreover, even if one takes LS based Regression approach, Valuing something based on just one specific ratio could be misleading as different Ratios reflect different components of the Valuation(Relative Valuation Approach).

Low Price to Book Value Vs ROE Regression(Reference Damodaran).Misuse of LSS based Regression to figure out Low to High Book Value. The determination of Low and High based on the Line of the Best-fit is itself faulty ! Again same typical Regression issue. Line of Best-fit itself could itself misleading in Real-World applications. So, what one figures out Low could be High if Line of Best Fit is changed !

 

There could be indeed better methodologies to incorporate different ratios dimensions of Valuation but that’s not the discussion of this paper here.

a)    Value in Real World is not the same as such Academic Theoretical Definition  as conventionally calculated in the Finance & Investment Literatures. The Definition of Value needs to be Redefined and Recalculated even in the Literature. This is the fundamental reason why Value has started underperforming over the time recently. So, we shall discuss on this in detail below.

 

 

This Theoretical Definition of Value is itself flawed and incomplete when we look at how Value works out in Real-World Markets for Investors.

How Investors figure out value in a stock traditionally is itself flawed based on conventional approaches.

There will always be some patterns in this world enough to mislead us and that could be due to chance.

 

 

 

 

 

 

 

 

 

31.  VALUE IN REAL SCIENTIFIC WORLD :REDEFINITION 

So, How this Valuation formulates in Real World. There is the difference between Academic Old Definition of Value and Real-World Scientific Value. Infact Academic definition of Value should be redefined.

 

Here is the brief formulation: Now taking the idea of Relative Valuations Multiple here…Valuation of a stock for an observer(investor) could be the different (Relative Valuation). Every Relative Multiple shows different component of Valuation. Selecting any particular multiple might be biased at times in real world finance.

The key drawback is that a firm can be overvalued and undervalued at the same time using different relative multiples. This reveals the key drawback of the process! How something is concluded to be Overvalued or Undervalued traditionally using any particular multiple e.g. Price to Book Value etc.  in Valued based Investment is both conceptually and statistically flawed !

Infact , Value depends on the optimal combination of different multiples with right weightage and randomness component. Not just that rather higher order dynamics of how those Value ratios, their trajectories etc. The different ratios have their own dynamics. Higher Order Analysis not just First Order Analysis also affect the Value.

That’s why I categorically repeat that the way Value has been traditionally and academically defined as Low Price to Book Value etc. is both foundationally as well as statistically flawed  in Real World.

Value is also relative on observer to observer. It’s NOT absolute !

 

)   This is just a basic mathematical simple framework to show but actually this is proprietary to decide the Value and would depend investor to investor relatively.

Further, it’s not just first order thing but also how these multiples behave over the time also determine the Value over the time.

 

So, What Value depends upon the combined approach for an investor. Conventional Academic Value itself needs Redefinition in  the Real World. Momentum, Quality etc. are the components of the True Consolidated Value not different entities from them in Real World. All these aspects/components determine the Value not just Cheapness on just low Price to Book Value multiple. It’s an Open -ended rather than a closed ended formula in every scenario.

Value  is scientifically determined by the Causal forces affecting human (investors’) Relative perceptions in real life and Cheapness of Price is just one of the many causal factors.

a)   Price to Book Value Multiple.

b)  Earning Multiples

c)    Debt Multiples

d)  Cash Flow Multiple

e)   Macro Economic Multiple

f)     Profitability & Growth Multiple

g)   Risk Multiple

h)  Cost Multiple

i)      Utility Multiple

j)      Momentum Multiple

k)   Sentimental Multiple

l)      Randomness

m)                       Others factors

This can be written as the statistical framework which remains proprietary here!

Further, I repeat time and again - it’s not just the first order rather how they have behaved dynamically over the time i. e. higher order would give better insight.

So, Value in Real World depends upon many Causal Multiples which actually affects the Business of a firm and Human(Investor behavior) scientifically  in a causal way as well as Randomness.

It’s not just Book to value Multiple. Infact, this traditional academic definition of Value is incomplete and needs to be redefined. Also, by this traditional definition, if something is cheap doesn’t mean all the investors would buy it. Cheapness is just one of the factors to buy. The IMPORTANT Point is We need to Understand How Value should be defined and how it works actually in the Real World.

One thing to Note that Momentum is also one of the Cause of Value . The reason being Momentum reflects the strength of Demand Supply forces being caused through Order Flow Imbalance. This force of demand supply also determines the value of something! Hence, contrary to traditional static absolute understanding of Value as different to Momentum, we have to understand it from the dynamic relative perspective that MomIentum is itself one of the Causes of Value and Vice versa !  Infact, it’s two ways : Value causes Momentum and Momentum causes Value. It’s essentially like Laws of Nature/Physics where Value is like Total Energy( including Potential) and Momentum is like Kinetic form.

So, all the other causal factors including momentum and randomness resultantly decide the Value.

Infact, Value is the function of many different causal factors including Momentum as well as unknown randomness !  This has its origin in the Human behaviors driving the Investors’ behavior at large coming from the science and laws of Nature & Human Behavior.

 

By the way, Scientifically using  Laws of Stability Equilibrium in Nature , there also exists Proprietary approach to calculate the fair Valuation Relatively using Geometrical Methods! We shall possibly discuss this later as much we can(due to proprietary nature) after talking about how Markets are linked to the Laws of Nature and Physics !

 

10.Causal Equation of Market

Let’s understand this in simple Newton’s Laws of motion like relationship in detail later but before that I write the Causal mechanism in the form of  Ornstein Uhlenbeck(OU) mean reversion Process where Stock price tends to revert to their true mean value over the time which is the state of equilibrium.

 This is Pure Scientific.

 

So in this equation, Net Effective Valuation at time t-

 cause the momentum at time t,  to move.

 

 

Valuation at time t is caused by the historical Momentum     as well.

                                         

 

 

But unlike interest rate dynamics here the mean reversion level keeps on changing over the time for a stock or market depending upon various valuation constituents.

                         

 

is the velocity at time t causing the momentum

is the Expected Stable Equilibrium State True Value Equivalent Calculated at time t. This will be calculated based as discussed above and results from the scientific dynamics of the forces.

What is extremely important to understand is that Even the Causal Equations have Error & Randomness Components which is the origin of Black Swan type events/ risks!

 

 

 

 

      

The typical parameters �,� and �, together with the initial condition �0, completely characterize the dynamics, and can be quickly characterized as follows, assuming � to be non-negative:

·        �b: "long term mean level". All future trajectories of � will evolve around a mean level b in the long run;

·        �a: "speed of reversion". � characterizes the velocity at which such trajectories will regroup around � in time;

·        �σ: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher � implies more randomness

The following derived quantity is also of interest,

·        �2/(2�) σ^2 / 2a: "long term variance". All future trajectories of � will regroup around the long term mean with such variance after a long time.

 

 

 

 

 

 

 

11.Scientific Background of Convexity & Randomized Approach in finance : Quantum Connection

 

Convexity :

_{More Gain than Pain from a Random Event. The performance curves outward, hence looks "convex". Anywhere where such asymmetry prevails, we can call it convex, otherwise we are in a concave position. The implication is that you are harmed much less by an error (or a variation) than you can benefit from it, you would welcome uncertainty in the long run.}

 

 

 

 

The Picture taken from Internet

 

Let’s look at the Science once again and then how Convexity, Randomized Tools like RCT, Algo wheel etc. Are Scientifically placed for Finance, Life etc..

As we know Feynman’s Path Integral approach has been the most important scientific result in Science which originated from Paul Dirac’s Principle of Least Action. So, as we know that Market is basically a  Quantum wave driven by the sum of many Human Brains, it would follow the Laws of Quantum world. Market would have uncertainty,randomness which could make it Unpredictable in some perspectives while they would follow some deterministic approach in other perspectives. These two aspects would co-exist simultaneously. We will have to discover that Determinism inside Randomenss/Uncertainty at some scale. So, rather than directly trying to predict the Uncertainty, one has to look at Uncertainty from indirect perspectives.  One has to find out the hidden Causal Mechanism inside Randomness. These all are being done while acknowledging the existence of Uncertainty at some scale. So, we may not directly predict Uncertainty at lower time scale or in individual perspective but we could definitely do that in global context. This is directly how Quantum world works. One might not predict the Behavior  of a Quantum entity like electron individually due to Uncertainty but definitely as a system. That’s where the role of Randomized based approach comes in. The most Predictive approach tries to predict at individual level but that’s not how the Quantum world works..They don’t acknowledge Uncertainty inherent in Nature. So, Scientifically one should find out global Certainty in Uncertainty/Randomness rather than trying to predict the Local Uncertainty! That’s what I had stated earlier that Randomness is NOT Randomly Random rather Deterministically Random .Randomenss also has cause. The discovery of that Determinism inside Randomness is the key to success in markets or real world or life.

Hence, One should conduct Random Trials in Causal way and discover the Determinism inside. It’s like Path of Least Action as in Feynman’s Path Integral or Paul Dirac’s Principle of Least Action . That Path inside Randomness is the key. And random trials should be based on some causal aspects not Randomly Random Trials always because for every Random Trial there would be some cost involved. One has to minimize that cost too. So, by following the Causal adjusted Random Trials, own can maximize Convexity. This is also linked to the law of energy. If the cost is not minimized, due to herd behavior of energy in markets, it could worsen leading to huge Tail risk.

 

So, it’s not Knowledge Vs Convexity rather Knowledge+ Convexity . Through Causal knowledge one can increase the Convexity by minimizing the cost of random trials. And this is also linked to Reproducibility Crisis where just taking random trials and selecting the best blindly could be biased as done in Backtesting etc.. ! This is because even the Best could behave randomly over the time.

 

That’s the clue to success : Discover the Determinism inside Randomness and follow the Path for Least Action . That’s based on Scientific Quantum Laws.

 

 

12. Causal -Adjusted Randomized Factor Approach

 

Now having understood the scientific concepts now deduce the approach for Causal Factor Investing. We see that Market participants keep on finding new and new factors and try to predict it using LS Regression type tools. They try to predict individual factors and try to find out the Ultimate complete set of Deterministic factors to predict the markets. That holy grail is utopian and scientifically could not possibly exist due to Quantum Nature of Market. The effort to make the Real World Deterministic which is fundamentally Quantum and having Uncertainty is against the Laws of Nature scientifically. Hence , that utopian dream of discovering the ultimate set of Deterministic Factors is against the Laws of Nature. It’s analogous to the ground breaking Godel Incompleteness Results in Markets where the effort to make market Deterministic by Complete set of Factors is like making Arithmetic a Complete Axiomatic system...

In other words, discovering the ultimate Holy Grail is like trying to assume the world as Deterministic, which is scientifically not possible and against the  Law of Quantum world.

So, the key is to follow Randomized Factor Approach. In this approach like Feynman’s Path Integral & Paul Dirac Principle of Least Action, we try to apply the same in Human Quantum World. This is where we try to discover the hidden Determinism inside Random Causal Factors. This way we explore Certainty inside inherent Uncertainty..That’s the key and analogous to the Law of Quantum world (Physics being just a subset of that ). RCT, Algo wheel are based on that principles but there are many other tools using which we try to study the Determinism inside Random Factors. So, like an electron, we may not predict exactly about every single factor deterministically rather obviously in collective sense. It’s like Quantum world  Interference, where we may not predict the trajectory of each electron, but definitely as a whole we can find some determinism that they would like around.

Hence, unlike traditional approach of selecting factors on one by one basis, the scientific approach should be to Randomize the Factors based on  some Causal understanding and create a Convex System.

For that, we can follow a rough algorithm as below

 

a)   Random Selection of  the Set of stocks based on the Causal based Deterministic and  Random Factors according to their Real-World Valuations as discussed above  .

b)  Allocation weights to them have to be dynamically arranged starting from some random component based on the historic performance, expected future scenarios in a well  diversified manner, putting   various constraints on the weight,while also looking at Tail Risk perspective idiosyncratically.

c)    The key is to discover the determinism out of Randomized approach over the time  follow the path of least action as a group on the principle the Nature works

 

d)  Allocate random weights to them at start to these selected Causal stocks 

e)   Some Predictive views based on Causality Factor Analysis could add up to weightages. Say for example if the has better risk-return profile expectation in future than , then even though randomly assigned weight .

f)     Put Constraints on the Weights to make it well diversified for example

     risk management point of view.

 

g)   Based on the discovered determinism inside their random trajectories over the time, keep on adjusting weights dynamically to each of them over the time.(for example)

There can be more sophisticated equations to update the weightage.

h)  The weights keep on adjusting dynamically based on determinism inside their randomness in real world.

g) Effectively, the Portfolio becomes relatively Convex to handle Black Swan Risk Effectively and revert to their equilibrium true state over the time….

 

The exact equations and functions to update weight would be changing in real world scenarios depending on the scenarios and one’s own risk-return objectives. Here it is just for demonstration perspective. The key aspect of Randomized Factor Approach is to Randomize the Factors selection in causal way and then discover the Determinism hidden inside over the time. This is unlike traditional approach where selected traditional associational factors are bet in deterministic way. But to acknowledge that the Holy grail of Completely Deterministic Factors won’t ever be found due to Quantum behavior of Markets. The key is to discover Causal Determinism inside Uncertainty/Randomness !

RCT/Algo wheel are some basic tools based on this approaches. The more advanced proprietary scientific tools based on this approach could be applied. It’s also based on the Convexity Principle to build the Portfolio dynamically over the time.

 

So, here True Risk is Black Swan Tail Risk, Normal Volatility should not be much taken as Risk rather they are the essential mechanism of the Markets ! As long as one is Convex and Prudent to Black Swan/Tail Risks, Intermediary Volatility should not be of much concern as according to the Laws of Nature and hence Markets, it would revert to the true Equilibrium Value state over the time.

 

 

 

 

 

 

 

 

13.  Summary & Conclusion

 

So, finally it’s time for summary now where I would briefly write the brief summary of the paper. I know some readers would like many views, others may disagree on certain points. It’s an open research paper. There is relativity kn thoughts. Different people can have different perspectives and all true.

We have to acknowledge and the part of highest intellectual maturity that duality in the form in of extreme opposites co-exist in Nature whose subset is finance, life etc. So, I have tried to unify two extremes of Randomness (Black Swan) & Causality; Predictability & Uncertainty in scientific way traversing the Quantum & Classical world. I’ve explained how both co-exist together and we need to appreciate it’s beauty.

Essentially Market driven by Human Quantum Behavior is a Quantum wave which is governed by certain laws of Energy and Nature. Black Swan which basically falls under the Unpredictability zone inherent in Nature,hence finance and life as well are essentially caused by sudden Energy shock, which is causal,but can’t be predicted as by definition Black Swan is Unpredictable(if it’s predictable, it’s not a Balck swan for the observer Relatively) . Black Swan (Caused by Randomness ) also follows the Law of Energy and Nature,which converges to its stable state level the time.

So,given the market being a Quantum human wave, it follows Certain laws of Nature & Energy scientifically. One need to understand that and hence manage Quantum type Uncertainty & Predictability both as Duality.

The clue to Financial Investment success is accepting the Certain Predictable components at the same time be convex to the Uncertainty. But to attain the maximum Convexity, one has to discover the Determinism inside Randomness. This causal approach would increase the Convexity by minimizing the cost.

For that rather tha finding the Deterministic set of Factors, one should follow Randomized Factor Approach . This is based on the Feynman & Paul Dirac Principle of Least Action. Where one should follow Causal adjusted Randomized factors and try to find out the Path for Least Action...Eventually AlgoWheel, RCT etc..are based on similar principles. But there can be more advanced approaches . Ultimately own has to discover Determinism inside Randomness, Certainty inside Uncertainty that exist inthe form of Duality in Nature,Life & Financial Markets.

Finance is essentially a form of Quantum Science involving Human world which must be studied in forward direction of time dynamically in Scientific & Causal Perspective(Randomness being part of that) rather than using backward looking conventional Statistical tools in static way . Role of Time Direction is quite important in Finance. One must understand this scientifically rather study finance as a statistical subject.

 

 

 

 

 

 

 

 

 

 References