The Bek Bernoulli Brochure—The RMR Essays


Table of Contents

Introduction

Christopher Bek Résumé

The Bernoulli Model

Risk Management Review Essays

Quotations

 


Risk Management Review Essays

Caption

Overview

Applying The Bernoulli Model

The Bernoulli Model

The Bernoulli Form

The Efficient Frontier

The Method of Moments

Scientific Management

 


Caption

There can be no other truth to take off from this—I think, therefore I exist—ie. the Cartesian cogito.  There we have the absolute truth of consciousness becoming aware of itself.  Every theory which takes man out of the moment in which he becomes aware of himself is, at its very beginning, a theory which confounds the truth, for outside the Cartesian cogito, all views are only probable, and a doctrine of probability which is not bound to a truth dissolves into thin air.  In order to describe the probable, you must have a firm hold on the true.  Therefore, before there can be any truth whatsoever, there must be an absolute truth; and this one is easily arrived at; it is on everyone’s doorstep; it is a matter of grasping it directly.

—Jean-Paul Sartre

 


Overview

Applying The Bernoulli Model describes the process of putting into play an executive risk management and forecasting system.  The Bernoulli Model re-cognizes the notion of wisdom—and argues that the world is on the cusp of a monumental paradigm shift due to the imminent fall of the authoritian model and the rise of portfolio theory in the practical incarnation of The Bernoulli Model.  The Bernoulli Form elucidates the notion of Platonic Forms and describes how a motley crew of Forms—including the Delphi, forecasting, integration, utility, optimization, efficiency and complementary—come together in the portfolio of Forms of The Bernoulli Model.  The Efficient Frontier examines the notions of God, option theory, portfolio theory, faith, reason and Arab math-finally arriving at the inescapable conclusion that all roads of sound decisionmaking lead to the efficient frontier.  The Method of Moments delineates dimensional deconstruction and reconstruction combined with fractal analysis as the fundamental method of riskmodeling employed by The Bernoulli Model.  Scientific Management follows the development of relativity from Archimedes to Einstein—and then takes a parallel line of reasoning in considering the development of scientific management and portfolio theory.

 


Applying The Bernoulli Model

Summary

Caption

Applying The Bernoulli Model

Essays

 

Summary

Applying The Bernoulli Model describes the process of putting into play an executive risk management and forecasting system. 

 

Caption

A successful business executive is a forecaster first—purchasing, production, marketing, pricing and organizing all follow.

—Peter Bernstein

 

Applying The Bernoulli Model

An Essay by Christopher Bek

 

Essay

Jewish religion stresses the fact that Scripture can be interpreted on many different levels.  Christ’s teachings encompassed themes that were already central to Jewish thought—for example, love and the importance of helping the unfortunate.  But he also taught the unorthodox thesis that Jewish law could be summarized in terms of loving God with one’s whole heart.  Christ sharply criticized those who made a great show of their holiness but who failed to show compassion—a theme again borrowed from the Hebrew prophets.  Muhammad (570-632) was a merchant in Mecca who became the central prophet and founder of Islam.  The term Islam derives from slam and means peace and surrender—namely, the peace that comes from surrendering to the will of God’s sovereignty.  Before Islam the religions of the Arabic world involved the worship of many gods—Allah being one of them.  Muhammad taught the worship of Allah as the only God, whom he identified as the same God worshipped by the Christians and the Jews.  And Muhammad also accepted the authenticity of both the Jewish prophets and Christ—as do his followers.  It is then accordingly clear that there is but one God.

 

The Bernoulli Model

The Bernoulli Model uses a top-down, strategic management approach to scientific management.  It combines the processes of forecasting, integrating and optimization.  The act of forecasting produces not only estimates of outcomes but also estimates of uncertainty surrounding outcomes.  In addition to the basic closed-form method of integration involving the two-moment normal distribution, The Bernoulli Model also employs the method of Monte Carlo simulation along with the four-moment the Camus Distribution in order to generate, capture and integrate the full spectrum of heterogeneously distributed forecasts of outcomes and the uncertainty surrounding outcomes.  Optimization algorithms search risk-reward space in order to determine the optimal set of decisions subject to Delphi constraints.  Closed-form optimization algorithms include linear programming while open-form methods include hill-climbing and genetic algorithms.

 

The Method of Prototyping

The method of prototyping involves developing a series of micro-models which eventually become macro-models.  In this case these micro-models are designed for use by the treasurer.  The Bernoulli Model further adds the Delphi program to more specifically define organizational values—utility theory to translate external values into internal values—Monte Carlo simulation to integrate heterogeneous risk components—the Camus Distribution to four-dimensionally represent risk—the Bernoulli moment vector for tracking forecasts—and an alternative hypothesis to serve as the loyal opposition to the null hypothesis.  The model also includes very stylish and highly-advanced Excel charts, VBA code and RoboHelp files.

 

Financial Indicators

A financial indicator designates a pointer of value—eg. VaR—Value at Risk, EaR—Earnings at Risk, CFaR—Cash Flow at Risk, UaR—Utils at Risk.  It is important to understand that each indicator includes a statistical distribution.  Here The Bernoulli Model utilizes the Bernoulli moment vector—each of which vector contains fourteen elements.  Financial indicators are the cornerstone of risk management.  These financial indicators emphasize what organizations hold of value.  The officers and directors designate the expected financial indicator values as well as the uncertainly or risk surrounding the chosen values.  Theses values are chosen by the officers and directors using the Delphi program.  The financial indicators expand on the definition of risk and reward.

 

Expanded Definition of Risk and Reward

Financial indicators are the objects by which risk and reward are measured.  Exposure (ie. Exp—M0) measures the initial exposure to change in value.  While the four-moment The Bernoulli Model provides an expanded definition of risk from the two-moment normal distribution to the four-moment Camus Distribution (ie. Mu—M1, SD—M2, Skew—M3, Kurt—M4) and fractal scaling (ie. Frac—M9) provide an expanded definition of risk.  Utility theory provides an example of an expanded definition of reward by changing external market values into internal Delphi values.  For example, a 100 percent external return (ie. Mu—M1) becomes a 50 percent internal return (ie. VaL—M5) while a –20 percent internal return becomes a –30 percent internal return.  While a 100 percent return is obviously desirable, undue emphasis on trying to achieve such a result may produce erratic outcomes and the missing out on more conservative opportunities.

 

Utility Theory

In 1905 Albert Einstein wrote one of the most profound documents ever written entitled Special Relativity Theory.  In 1952 Harry Markowitz wrote one of the most profound documents ever written entitled Portfolio Selection Theory.  In 1738 Daniel Bernoulli (1700-82) wrote one of the most profound documents ever written entitled Utility Theory—the full name of the theory being—The Exposition of a New Theory on the Measurement of Risk.  The central theme of Utility Theory being that the value of an asset is the utility it yields rather than its market price.  His paper delineates the all-pervasive relationship between empirical measurement and gut feel.  The utility function converts external, market returns into internal, Delphi returns.

 

The Bernoulli Moment Vector

The Bernoulli moment vector tracks risk and return forecasts via a fourteen-element vector.  The Markowitz Model uses the mean to represent the forecast or reward and the standard deviation to represent the dispersion or risk—thus laying the groundwork for risk-reward efficiency analysis.  The method of moments is a simple procedure for estimating the statistical moments of a distribution.  The mean is the first moment of a distribution and is calculated as the average value—and the standard deviation is the second moment and is calculated as the average deviation about the mean.  The Bernoulli Model also employs an expansion on the method of moments with the Bernoulli moment vector relating to the aggregate portfolio distribution.  The zero moment in the Bernoulli moment vector represents exposure.  The Camus Distribution represents the first four moments.  The fifth moment is VaL and represents a utilitarian translation of reward and thus an expanded definition of reward.  The sixth moment is VaR and represents the confidence level and thus an expanded definition of risk.

 

The Complementary Principle

Niels Bohr is one of the founding fathers of quantum theory who also defined the complementary principle as the coexistence of two necessary and seemingly incompatible descriptions of the same phenomenon.  One of the first realizations dates back to 1637 when Descartes revealed that algebra and geometry are the same thing—ie. analytic geometry.  The Bernoulli Model allows for the separation of the null and alternative hypothesis.  This ability to compare paradigms represents an invaluable feature of The Bernoulli Model.  The Bernoulli moment vector represents the tabular depiction of the Bernoulli portfolio while the Excel charts represent the graphical form of the Bernoulli portfolio.

 

The Delphi Program

The Delphi program is the overriding guidance system for The Bernoulli Model.  It employs the iterative Delphi method designed to draw out fundamental values from officers and directors.  The purpose of the Delphi process is to streamline decisionmaking for all concerned.  The Delphi program is named after the Socratic inscription—Know Thyself—at the oracle at Delphi in ancient Greece.  The primary Delphi value pertains to the confidence level and value for allowable downside risk exposure of the portfolio distribution—using a financial indicator like VaR.  Secondary Delphi value pertains to the utility translation function and risk-reward efficiency analysis.  

 

The Actuarial Valuation Process

The actuarial valuation worksheet shows the progression of the financial indicators through the valuation process.  The actuarial valuation worksheet shows the end result of the technical development.  The worksheet contains the Bernoulli moment vector for both the components and the portfolio.  It also contains advanced Excel charts that are broken down into the null configuration and the alternative configuration.  Anything that is not made clear from the actuarial valuation worksheet can be found in the technical analysis worksheet.  This worksheet shows the development of the actuarial valuation process.

 

The Treasurers’ Perspective

The Bernoulli Model is a top-down strategic management, forecasting and risk management system that is mathematically accessible to executives.  It is designed for use by the treasurer showing the actuarial valuation Excel worksheet illustrating a storyboard that demarcates the same six Excel charts for all organizational financial indicators used in the actuarial valuation process.  The worksheet also presents both the null and alternative valuation parameters and the null and alternative Bernoulli moment vectors.  The actuarial valuation worksheet also includes valuation parameters as well as the Bernoulli moment vectors.  All of this leads to a brand new look at scientific management for the treasurer.

 

Conclusion

Peter Bernstein once said that risk is no longer bad news—rather it is a harbinger of opportunity to be harnessed for our benefit.  This essay describes the process of putting into play an executive risk management, decisionmaking and forecasting system.  Sir James Jeans once said that God is a mathematician.  The notion of God as a mathematician is in fact consistent with the idea that there is only on God.

 


The Bernoulli Model

Summary

Caption

The Bernoulli Model

Essays

 


Summary

The Bernoulli Model re-cognizes the notion of wisdom—and argues that the world is on the cusp of a monumental paradigm shift due to the imminent fall of the authoritian model and the rise of portfolio theory in the practical incarnation of The Bernoulli Model.

 


Caption

The time has come, the Walrus said, to speak of many things.

—Lewis Carroll

 


The Bernoulli Model

An Essay by Christopher Bek

 

Essay

The word philosophy comes from ancient Greece and is defined as the love of wisdom.  In a recent hockey game between the Calgary Flames and the Edmonton Oilers—the so-called Battle of Alberta—the twenty-one year-old millionaire and Alberta native Mike Comrie appeared to be hit with a high stick by the twenty-three year-old millionaire and Alberta native Derek Morris, thus giving the Oilers a two-minute powerplay.  But the replay showed that Comrie had in fact faked being hit.  Upon seeing this excellent deception, the venerable Canadian sportscaster Jim Hughson congratulated Comrie for being wise beyond his years.  In other words, Hughson had, in no uncertain terms, splained to the children that, in Canada, wisdom and lying are the very same thing.

 

Bringing Down the Government

The Czech Václav Havel once said that corruption begins when people start saying one thing and thinking another.  In his essay The Power of the Powerless Havel asserts that people can bring down a dictatorial government nonviolently by simply living in truth.  John Maynard Keynes was a British economist whose ideas continue to profoundly influence government policy to this day.  Keynes amassed a personal fortune during the 1920s by speculating on the fluctuations of currency exchange rates.  He addressed the problem of boom-bust cycles that constantly plague capitalism by arguing that government should increase or decrease spending in accordance with cyclical fluctuations.

 

Fair?  Foul?  Whatever

Keynes is most noted for the prophecy made in 1930 as he looked beyond the foreboding presence of the looming Great Depression—When the accumulation of wealth is no longer of high social importance, there will be great changes in the code of morals.  We shall be able to rid ourselves of many of the pseudo-moral principles which have hag-ridden us for two hundred years, by which we have exalted some of the most distasteful human qualities into the position of the highest virtues.  The love of money as a possession—as distinguished from the love of money as a means to the enjoyments and realities of life—will be recognized for what it is, a somewhat disgusting morbidity, one of those semi-criminal, semi-pathological propensities which one hands over with a shudder to the specialists in mental disease.  But beware!  The time for all this is not yet.  For at least another hundred years we must pretend to ourselves and to everyone that fair is foul and foul is fair—for foul is useful and fair is not.  Avarice, usury and precaution must be our gods for a little longer still.  For only they can lead us out of the tunnel of economic necessity into daylight.

 

Playing the Dad Card

Einstein once said that God had punished him for his contempt of authority by making him an authority himself.  During the 1960s the American psychologist Stanley Milgram performed a remarkable experiment for testing the obedience to authority of one thousand subjects.  An authoritian figure ordered each of the subjects to administer increasingly painful electrical shocks to a learner every time the learner either failed to answer or answered a question incorrectly.  The learner, who could be heard and not seen was, in fact, not actually given the shocks.  The punishment began at 15 volts and increased in 15-volt increments to 450 volts.  In spite of the fact that the learner would scream in pain, beg for mercy, and finally fall silent at 330 volts—a full two-thirds of the subjects delivered the final punishment of 450 volts.

 

Romancing the Moon

Wittgenstein once said that philosophy is the battle against the bewitchment of our intelligence by the means of our language.  Just as numbers are a useful fiction leading toward eternal verities in mathematics—so too are words the-finger-pointing-at-the-moon when endeavoring to understand ultimate philosophical truths.  We could certainly all agree that society would not have come this far without some form of auth­ority to guide it.  But the question is—At what point does the authority of government cease to guide us forward and, instead, choose itself over the people?  Existentialism is the philosophy which asserts that morality must be determined inwardly rather than from external authority.  Consider that the Freudian cognitive model makes the reality-based ego the decisionmaker who must choose between the internal values of the inward self or soul and the external authority of the superego.  Gandhi used to speak disparagingly about systems so perfect that no one would have to be good.  But the goal is not to make goodness obsolete—but to create systems which align internal and external values.

 

The Authoritian Model

Einstein once said that there is no more commonplace statement to make than the world in which we live is a four-dimensional spacetime continuum.  Consider for a moment that both morality and risk management are similarly exercises in making decisions in the face of uncertainty.  From this it follows that we can only know whether decisions are correct or not retrospectively.  But the very definition of authority is that it is able to make determinations of right and wrong at any time.  And the authoritian strategy is dead simple—bully the ego into making the easy, low-risk, low-reward, short-term decision of obeying authoritarian rules—rather than allowing the ego time to reflect on the values of the soul and possibly make the higher-risk and ontologically higher-reward decision.  In essence, the authoritarian model forces us to remain in the three-dimensional world because it can neither feed off us nor control us in the four-dimensional world.

 

The Markowitz Model

Harry Markowitz developed portfolio theory in 1952 as a way of constructing optimal portfolios that maximize reward for given levels of risk.  Markowitz forever linked reward with risk in exactly the same way that Einstein linked space with time in that both the expected outcome and the attendant uncertainty of outcome are required to complete the picture.  His approach employs matrix algebra to aggregate risk, and then uses linear programming to determine optimally efficient portfolio allocations.  Just as Galileo and Descartes laid down the fundamentals of the modern scientific method for solving problems—so too did Markowitz lay down the fundamentals of the modern scientific method of four-dimensionally bringing together a varied set of uncertain elements.  But rather than seeing portfolio theory for the profoundly important roadmap that it is—the individual components of the model have been incessantly and pedantically attacked by PhDs and other authorities—like the legions of PhDs at Enron before the company finally collapsed under the weight of its own PhDness.

 

The Bernoulli Model

Newton once said that if he had seen further than others, it was because he stood on the shoulders of giants.  In 1670 both Newton and Leibnitz formulated versions of calculus—that is, the mathematics of motion.  The Bernoulli brothers, John and James, picked up on calculus, spread it through much of Europe, and then set the roadmap for efficiency analysis by finding the curve for which a bead could be slide down in the shortest time.  John’s son Daniel set the roadmap for utility theory based on the idea that an asset’s value is determined by the utility it yields rather than its market price.  No less than eight Bernoulli’s made significant contributions to mathematics.  The Bernoulli Model is founded on the shoulders of giants like Archimedes, Galileo, Descartes, Newton, Leibnitz, Einstein, Markowitz and the Bernoulli’s.  It is singularly directed towards establishing balance and efficiency for institutions and individuals within institutions.

 

To Infinity and Beyond

The word stochastic comes from ancient Greece and is defined as skillful aiming.  The Greek Plato once said that a just society would only be possible once philosophers became kings and kings became philosophers.  The Bernoulli Model makes the president the focalpoint—and provides a coherent enterprise-wide view.  It encourages a priori definitions of values, opinions and data specifications so as to facilitate actuarial efficiency—and interactive, nonauthoritian communication between the board and the executives, and the executives and the operations.  The model’s a priori conception offers expandability along a multitude of dimensions including forecasting, risk-modeling, efficiency analysis and utilization—as well as offering full accountability and comparability for all risk factors.  The Bernoulli Model also employs the state-of-the-art, four-moment Camus distribution—which in turn sets the roadmap for the realization of the vast untapped potential of simulation-based optimization.

 

Conclusion

FS Northrop once said that if one makes a false or superficial beginning, no matter how rigorous the methods that follow, the initial error will never be corrected.  It is said that addiction stems from the inability to conceive of the future.  Simply put, the addict never knows his own soul.  Alberta is the richest province in the richest country in the world—yet a remarkable portion of the provincial revenue comes from video lottery terminal machines—ie. VLT machines.  As the wise and venerable premier of Alberta, Ralph Klein, responded when asked the question as to when he would leave government—Once the government is operating like a well-oiled machine.

 


The Bernoulli Form

Summary

Caption

The Bernoulli Form

Essays

 


Summary

The Bernoulli Form elucidates the notion of Platonic Forms and describes how a motley crew of Forms—including the Delphi, forecasting, integration, utility, optimization, efficiency and complementary—come together in the portfolio of Forms of The Bernoulli Model.

 


Caption

In 1952 a young graduate student named Harry Markowitz studying operations research demonstrated mathematically why putting all your eggs in one basket is an unacceptable strategy and why optimal diversification is the best one can do.  His revelation touched off an intellectual movement that has revolutionized Wall Street, corporate finance and decisionmaking of all kinds.  Its effects are still being felt today.

—Peter Bernstein

 


The Bernoulli Form

An Essay by Christopher Bek

 

Essay

On 14 December 1900 Max Planck (1858-1947) told his son that he had just made a discovery as important as that of Newton.  Planck revealed why we are able to stand so close to a fire without being overwhelmed by radiation.  He realized the fact that energy is transferred in discrete packets or quanta defined by Planck’s constant puts a size restriction on escaping energy units thus causing a traffic jam.  In 1925 Planck’s constant formed the basis of quantum theory—which is the natural law of matter and explains the periodic table and is the foundation of electronics, chemistry, biology and medical science.  In 1905 Albert Einstein (1879-1955) produced three papers—The Photoelectric Effect, which applies Planck’s quantum concept to light—Brownian Motion, which delineates the stochastic process and is the basis of all riskmodeling—and Special Relativity, which is the natural law of spacetime.  In 1906 Planck wrote to the unknown Einstein and acknowledged the greatness of his discoveries.  In 1915 Einstein adapted the curved geometry of Bernhard Riemann (1826-66) as the underlying a priori Form for general relativity.  Special relativity refurbished Newtonian physics in respect of uniformly moving bodies traveling in straight lines—and general relativity upgraded special relativity to account for bodies traveling at varying speeds along curved lines.  On 28 May 1919 Sir Arthur Eddington led an expedition to the island of Principe off the coast of Africa to photograph an eclipse of the sun.  Analysis revealed a warping of spacetime consistent with general relativity thereby providing a posteriori validation.  Planck stayed up all night awaiting the results while Einstein slept like a baby.  When asked what he would have done had the results not confirmed his theory, Einstein responded by saying—Nothing, for the good Lord must have errored.

 

Platonic Forms

The Greek Plato’s (427-347 BC) theories of knowledge and Forms holds that true or a priori knowledge must be certain and infallible, and it must be of real objects or Forms.  Thales and Pythagoras laid the foundation for Plato by founding geometry as the first mathematical discipline.  Mathematics is the systematic treatment of Forms and relationships between Forms.  It is the science of drawing conclusions and is the primordial foundation of all other science.  The Greeks synthesized elements from the Babylonians and Egyptians in developing the concepts of proofs, axioms and logical structure of definitions—which is mathematics—which when combined with a posteriori validation allows us to arrive at a priori knowledge.  While Thales introduced geometry, it was Pythagoras who first proved the Pythagorean theorem which establishes a priori knowledge that the square of the hypotenuse of a right-angle triangle is equal to the sum of the squares of the two sides.  Both Einstein’s relativity in 1905 and my theory of one in 2001 make use of the Pythagorean theorem as their underlying a priori Form.  Relativity derives its a posteriori validation from the 1887 Michelson and Morley experiment while the theory of one gets its a posteriori validation from the 1982 Aspect experiment.

 

Existence and Essence

The term a priori refers to a four-dimensional mathematical essence while a posteriori refers to a three-dimensional commonsense existence.  Essence is the true kernel of a thing while existence simply refers to the sheer fact that a thing is.  The soul is an essence while the ego merely exists.  William Barrett wrote in his 1958 book Irrational Man that the history of Western philosophy has been one long conflict between existentialism and essentialism.  Jean-Paul Sartre (1905-80) defined existentialism as the philosophy for which existence precedes essence.  Conversely, essentialism asserts that essence precedes existence.  The problem is that precedence is a temporal operator and essence is outside time—meaning that the notion of precedence here is meaningless.  It is the age-old problem of the chicken and the egg.  As a fundamental attitude, The Bernoulli Model is existential—but based on a portfolio of empty Forms—with the faith being that the application of the model will animate the Forms thus realizing the essence of model.

 

The Delphi

Existentialism is based on the self-verifying Form of the Cartesian cogito—I think, therefore I exist.  The Bernoulli Model employs the self-verifying Form of the Delphi method which is an iterative process intended to draw out executive values—and is named after the Socratic inscription on the oracle at Delphi—Know thyself—which is of course equivalent to the Cartesian cogito and also equates to the objective function from operations research.  The basic Delphi value pertains to allowable downside risk exposure for the portfolio distribution.

 

Forecasting

The statistical distribution is one of the most beautiful Forms for the reason that it represents both the forecast of outcomes as well as the expected uncertainty.  Advanced forms of forecasting of The Bernoulli Model include intertemporal riskmodeling—which is able to accurately represent time-series data like energy prices and foreign exchange rates characterized by contemporaneous and intertemporal dependencies.  The approach deconstructs historical data into signal, wave and noise—each of which is then forecast separately.

 

Integration

Integration is the process of aggregating or bringing together forecasts of outcomes and uncertainty.  The closed-form method of integration involves matrix algebra and applies strictly to the two-moment normal distribution.  The Bernoulli Model also employs the open-form method of Monte Carlo simulation with the four-moment Camus distribution in order to capture and integrate the full spectrum of heterogeneously distributed forecasts.

 

Utility

Daniel Bernoulli (1700-82) founded utility theory by writing a paper entitled Exposition of a New Theory on the Measurement of Risk—with the theme being that the value of an asset is determined by the utility it yields rather than its market price.  His paper, one of the most profound ever written, delineates the all-pervasive relationship between empirical measurement and gut feel.  The Bernoulli Model employs utility theory by adjusting market returns to more accurately represent internal values.

 

Optimization

Optimization is part of operations research that originated in World War II when militaries needed to allocate and deliver scarce resources to operations.  Optimization algorithms search either cost-function or risk-reward space to determine the optimal value for the objective function subject to Delphi constraints such as allowable downside risk exposure.  Local search algorithms include linear programming and hill-climbing algorithms while global search algorithms include genetic algorithms.

 

Efficiency

In risk-reward space the process of optimization is carried-out for every level of risk with the result being the construction of the efficient frontier.  A similar process in cost-function space is known as data envelopment analysis.  The Bernoulli brothers, James (1654-1705) and John (1667-1748), set the roadmap for efficiency analysis by finding the curve for which a bead could be slide down in the shortest time.  The efficient frontier has come to form the bedrock of portfolio theory since its introduction in 1952 by Harry Markowitz.

 

Complementary

Niels Bohr (1885-1962) defined the complementary principle as the coexistence of two necessary and seemingly incompatible descriptions of the same phenomenon.  One of its first realizations occurred in 1637 when Descartes revealed that algebra and geometry are the same thing.  In 1860 Maxwell revealed that electricity and magnetism are the same thing.  In 1915 Einstein revealed that gravity and inertia are the same thing.  The ability to contrast paradigms presents the invaluable feature of the complementary perspective.

 

The Agency Problem

The agency problem is the pervasive predicament whereby agents select against organizations.  The first realization arose between tenant farmers and landowners.  A business manager who invests in marginal projects so he can reaps the benefits of being the manager of a larger portfolio is selecting against the organization.  The risk measuring concept of VaR originated because traders were playing the game of heads-I-win-tails-you-lose and exposing organizations to huge risks.  When gambles went south the traders simply moved on.  The problem now is that traders are gaming VaR.  Owing to its mathematical basis—The Bernoulli Model is made virtually impenetrable to the agency problem.

 

Conclusion

The Harvard Business Review publication began in 1922 with the intention of connecting fundamental economic and business theory with everyday executive experience.  The Bernoulli Model represents a systematic realization of that very mandate.  The approach frees executives from political gridlock and offers the ability to either skim the surface of decision analysis or drill-down and examine the inner workings of decisions and asset valuations.  It affords a comprehensive overview and provides unequivocal confidence allowing executives to sleep like babies knowing what Einstein knew—when the math is good the math is good.

 


The Efficient Frontier

Summary

Caption

The Efficient Frontier

Essays

 


Summary

The Efficient Frontier examines the notions of God, option theory, portfolio theory, faith, reason and Arab math-finally arriving at the inescapable conclusion that all roads of sound decisionmaking lead to the efficient frontier.

 


Caption

God is a mathematician

—Sir James Jeans

 


The Efficient Frontier

An Essay by Christopher Bek

 

Essay

The French mathematician and philosopher Blaise Pascal (1623-62) originated option theory with his famous wager regarding the questions of existence and ultimate nature of God.  His argument came during the Renaissance in response to those unwilling to believe in God strictly on faith and authority.  Pascal argued that living a simple life which seeks to understand God represents the option premium which then allows for the possibility of salvation should it turn out that God does exist.  Some critics have argued that God might well reserve a special place in Hell for those who believe in Him on the basis of Pascal’s wager.  But in fact the exact opposite is true.  Those who believe in God strictly of the basis of faith are setting themselves up for failure for the reason that their conception of God is based on a static snapshot that is, by definition, not subject to reason.  The Devil is the one who seeks out those who blindly follow.  A true God most certainly wants to be constantly challenged by both faith and reason.  Kevin Spacey tells us in the 1996 movie The Usual Suspects that the greatest trick the Devil ever pulled was convincing the world he doesn’t exist.  And now we know the second greatest trick the Devil ever pulled was convincing the world we can know God by faith alone.

 

Option Theory

Option theory is the decisionmaking methodology whereby decisions to invest or not are deferred with the purchase of options.  For example, a drug company might enter into a relationship with a university for the purpose of gaining access to research projects.  Analyzing the strategic value of such projects is difficult as the result of the prolonged developmental phase of pharmaceuticals as well as the complexity involved in predicting the future market.  Relationships are structured such that the company pays an up-front premium followed by a series of progress premiums until the company chooses to either purchase the research at an agreed-upon price—or discontinue the progress premiums, thereby forfeiting any future option-to-purchase rights.

 

The Heart of Risk

Risk analysis originated in 1654 when Pascal and another mathematician named Pierre de Fermat solved the problem of how to divide the stakes for an incomplete game of chance when one player is ahead.  The problem had confounded mathematicians since it was posed two hundred years earlier by a monk named Luca Paccioli who coincidently also introduced double-entry bookkeeping.  Their discovery of the theory of probability made it possible for the first time to make decisions and forecasts based on mathematics.  Just like questions of the existence and nature of God, the serious study of risk originated during the Renaissance when people broke free from authoritian constraints and began subjecting long-held beliefs to philosophic and scientific enquiry.

 

Greek Mathematics

The Greek Thales (625-546 BC) launched philosophy and mathematics after having amassed a fortune by first forecasting bumper olive crops and then purchasing options on the usage of olive presses.  According to Plato (427-347 BC) true or a priori knowledge must be certain and infallible and it must be of real objects or Forms.  Mathematics is thus the systematic treatment of Forms and relationships between Forms.  It is the science of drawing conclusions and is the primordial foundation of all other science.  Saint Augustine (354-430) carried forward Greek thought from the failing classical world to the emerging medieval, Christian world—a project that came to be known as the medieval synthesis.  For twelve hundred years the flame of philosophy and science lit by Augustine burned ever so lowly under the agonizing oppression of the Church.  Copernicus published On the Revolution of Celestial Orbs in 1543 mathematically proving the theory of heliocentricity.  And then by inventing and using of the telescope, Galileo (1564-1642) was able to provide the empirical validation of heliocentricity—for which the Church sentenced him to life in prison.  The French philosopher and mathematician René Descartes (1596-1650) shared Galileo’s views and envisioned the masterful strategy of presenting these revolutionary ideas to the Church in such a way that the Church believed the ideas were their own.  His heroic plan succeeded and the philosophic and scientific Renaissance of the seventeenth century was born.

 

Arab Algebra

While the Church was jumping up and down on everyone’s head for over a millennium, Arab mathematicians like Muhammad al-Khwârizmî (780-850) were carrying the ball in founding algebra and algorithms.  An algorithm is the procedural method for calculating and drawing conclusions with Arabic numerals and the decimal notation.  Al-Khwârizmî served as librarian at the court of Caliph al-Mamun and as astronomer at the Baghdâd observatory.  Both the terms algebra and algorithm stem from the God, Allah.  According to Arab philosophy, mathematics is the way God’s mind works.  The Arabs believe that by understanding mathematics they are comprehending the mind of God.  In fact the core of their religion lies with the belief that people must submit to the will of God—meaning mathematical arguments.

 

Analytic Geometry

The Latin version of al-Khwârizmî’s work is responsible for a great deal of the mathematical knowledge that resurfaced during the Renaissance.  In fact, the notion that mathematics and God are the same thing was adapted as the foundation for the Renaissance by thinkers like Descartes, Pascal, Fermat, Newton, Locke and Berkeley.  Then, in what John Stuart Mill called the single greatest advance in the history of science, Descartes conceived analytic geometry by synthesizing Greek geometry with Arab algebra.  The significance of this founding of modern mathematics is best understood in light of the fact that mathematicians from that point forward had two complimentary and fundamentally different ways of viewing the same Forms.  Einstein first introduced relativity theory in 1905 as a simple set of algebraic equations, yet the theory was ignored until four years later when Minkowski presented a geometric view of relativity as characterized by the four-dimensional spacetime continuum.

 

The Cartesian Method

In addition to founding modern mathematics, Descartes also found modern philosophy by tearing down the medieval house of knowledge and building again from the ground up.  By employing the method of radical doubt, Descartes asked the question—What do I know for certain?—to which he concluded that he certainly knew of his own existence—cogito, ergo sum—I think, therefore I exist.  Based on the natural light of reason, Descartes formulated his famous Cartesian method which is—Only accept clear and distinct ideas as true—Divide problems into as many parts as necessary—Order thoughts from simple to complex—Check thoroughly for oversights—And rehearse, examine and test arguments over and over until they can be grasped with a single act of intuition or faith.  Descartes rightly believed his method would guarantee certain and infallible knowledge.  Initially, one faithfully or intuitively senses truth, which is followed up by constructing rational arguments and then intuitively capturing completed arguments.  In other words, faith leads us to reason and then reason leads us back to faith.

 

The Markowitz Model

In 1952 a twenty-five year-old graduate student named Harry Markowitz studying operations research at the University of Chicago strung together three algorithms—forecasting, integration and optimization—ie. method of moments, matrix algebra and linear programming—in developing portfolio theory as a way of constructing optimally efficient portfolios that maximize reward for a given level of risk—with the efficient frontier being constructed by optimizing for all levels of risk.

 

The Bernoulli Model

In 1690 the Bernoulli brothers set the roadmap for efficiency analysis by finding the curve for which a bead could be slide down in the shortest time.  The Bernoulli Model upgrades the three algorithms of The Markowitz Model—forecasting, integration and optimization—with—intertemporal riskmodeling and decision trees, Monte Carlo simulation and the Camus distribution, and genetic and hill-climbing algorithms—and adds the Delphi process, utility theory and the complimentary principle.  The approach essentially provides an efficiency workshop for realizing the vast potential of The Cartesian Method.

 

The Orb of Efficiency

The Delphi process identifies first-order values that rise above cost-benefit such as allowable downside risk exposure.  The second-order objective is to ensure portfolio risk-reward efficiency.  The efficient frontier represents the best that one can do in terms of maximizing expected reward for each level of expected risk.  It depicts the panoramic fruition of the highest forecasting and decisionmaking intelligence for the organizational portfolio.  And while the end result is sufficient enough reason for conducting the exercise in the first place, the process of going through the analysis is often worthwhile in and of itself.

 

Conclusion

Starting from the realization that the very definition of the word religion means a reconnection with reality—we know that most organizations, religious or otherwise, rest on unchallenged preconceptions.  The whole point of applying option theory and following through on the efficient frontier is a recognition of the fact that not only situations but our conception of situations changes as we go.  To think like a mathematician then is to—as Socrates rightly asserted—follow the argument wherever it leads.

 


The Method of Moments

Summary

Caption

The Method of Moments

Essays

 


Summary

The Method of Moments delineates dimensional deconstruction and reconstruction combined with fractal analysis as the fundamental method of riskmodeling employed by The Bernoulli Model.

 


Caption

There is no more commonplace statement than the world in which we live is a four-dimensional spacetime continuum.

—Albert Einstein

 


The Method of Moments

An Essay by Christopher Bek

 

Essay

In 1975 the Polish mathematician Benoit Mandelbrot posed the question—How long is the coastline of Britain?  Appealing to relativity, Mandelbrot pointed out that it depends on one’s perspective.  From space the coastline is shorter than to someone walking because on foot the observer is exposed to greater detail and must travel farther.  According to Mandelbrot, when the shape of each pebble is taken into account, the coastline turns out to have infinite length.  He proposed a system for measuring irregular shapes by moving beyond integer dimensions to the seemingly absurd world of fractional dimensions.  Mandelbrot used a simple procedure involving the counting of circles to calculate the fractal dimensionality.  The coastline of Britain has a fractal dimension of 1.58 while the rugged Norwegian coastline is 1.70.  Coastlines fall in between one-dimensional lines and two-dimensional surfaces.  In the three-dimensional world the fractal dimension of earth’s surface is 2.12 compared with the more convoluted topology of Mars estimated to be 2.43.

 

Self-Similarity

A fractal is a mathematical Form having the property of self-similarity in that any portion can be viewed as a reduced scale replica of the whole.  Fractal structures exist pervasively throughout nature because theirs is the most stable and error tolerant.  The fractality of clouds is evidenced by the fact that they look the same from a distance as they do up close.  Mountains, snowflakes, lightning, galaxy clusters, earthquakes and broccoli are just a few of the naturally occurring phenomena that exhibit fractal qualities.  The power of fractal analysis lies in its ability to capitalizes on self-similarity across scale by locating an eerie kind of order lurking beneath seemingly chaotic surfaces.  And not only is fractal analysis scalable across applications but, owing to its mathematical foundation, it is also portable between applications.

 

Lost Moments in Time

While the coastline conundrum involves using fractal analysis to measure the complexity of geometrical shapes, the British hydrologist Harold Hurst (1900-78) employed fractal risk analysis in managing the Nile river dam from 1925 to 1950—with the goal being the formulation of an optimal water discharge policy aimed at balancing overflow risk with the risk of insufficient reserves—a job description not unlike that of a treasurer’s.  Hurst initially assumed the influx of water followed a random walk—although he abandoned that assumption in favor of a more robust fractal process.  A random walk or Brownian motion is a statistical process that has no memory and is represented by the normal distribution.  The fractal process is a superset of the random walk where the fractal dimension ranges from zero to one—with a value of 0.5 being the normal distribution.  Hurst’s work on the project led him to examine 900 years worth of records the flood-weary Egyptians kept.  Capitalizing on self-similarity, he analyzed data under all available time-scales from phenomena including river and lake levels, sun-spots and tree-rings in calculating a fractal dimensionality of 0.75.

 

Normal and Singularistic Science

In normal science a singularity is a breakdown in spacetime such that the laws of physics no longer apply.  Typical examples of singularities include the big bang, black holes and one divided by zero.  What physicists like Stephen Hawking who developed the concept of singularities failed to realize is that a breakdown in spacetime is just another way of saying a boundary of spacetime.  Thomas Kuhn (1922-96) was a physicist and historian concerned with the sociology of scientific change.  In his 1962 book The Structure of Scientific Revolutions he defines the term paradigm shift as a transformation taking place beyond the grasp of normal cognitive abilities.  Scientists apply normal scientific methods within a paradigm until the paradigm weakens and a shift occurs.  Most people eat up normal science with a big spoon, but do everything possible to avoid the intense metaphysical pain of paradigm shifts.  Hawking once said that a singularity is a disaster for science.  But what he should have said is that a singularity is a disaster for normal science—but normal for singularistic science.

 

Normal and Singularistic Distributions

The range of the fractal process maps isomorphically to a family of distributions known as the fractal or stable Paretian—named after Vilfredo Pareto (1848-1923).  There are explicit expressions for three fractal distributions—the Bernoulli (ie. coin toss), the normal and the Cauchy—corresponding to a fractal exponent of zero, 0.5 and one respectively.  The Bernoulli, named after James Bernoulli (1654-1705), converges to the normal distribution when the number of coins becomes sufficiently large.  The Cauchy, named after Augustine Cauchy (1789-1857), is interesting in that it possesses undefined moments—thus making it singularistic.  The first four moments of a statistical distribution are the mean, standard deviation, skewness and kurtosis—with kurtosis being a measure of both pointedness and length of tails.  The extremely long tails of the Cauchy give rise to its undefined moments.  The mean never converges because a value sampled from the extreme of the tails shifts any previously established mean.  The Cauchy is related to the normal in that it is a normal divided by a normal.  And one can easily see this in Excel by simulating a normal sample with the formula =normsinv(rand()).  If the simulated denominator is very close to the mean of zero—then the value of the Cauchy shoots off to the moon.

 

The Perfect Actor

I developed the four-moment Camus distribution—named after Albert Camus (1913-60) for his desire to be the perfect actor—as a one-size-fits-all distribution to model the full range of the fractal process.  So whereas the basic Bernoulli has a kurtosis of zero, the normal has a kurtosis of three and the Cauchy has infinite kurtosis—the Camus with a fractal dimensionality of 0.75 has a kurtosis of six.  What the Camus does is interpolate between the Bernoulli and the normal or the normal and the Cauchy—depending on the fractality.  The normal distribution with its fractality of 0.5 translates into scaling according to the square-root-of-time.  Going from a one-month valuation period to a one-year valuation period under a normal assumption results in a scaling factor of 3.5—ie. 12^0.5—while a similar calculation with a Camus distribution and a kurtosis of six produces a scaling factor of 6.4—ie. 12^0.75.  The rationale being that with higher kurtosis comes the greater potential for larger jumps.

 

Intertemporal Riskmodeling

The Hurst Model example involves components characterized by intertemporal dependencies and The Markowitz Model represents portfolio analysis involving components characterized by contemporaneous dependencies.  The Bernoulli Model is a superset of both that includes intertemporal riskmodeling as an approach representing data characterized by both intertemporal and contemporaneous dependencies—such as energy prices and foreign exchange rates.  The word stochastic comes from ancient Greece and is defined as skillful aiming.  While the basic stochastic process is the random walk, intertemporal riskmodeling expands along a multitude of moments and dimensions.  The random walk process bifurcates into the Camus distribution and a mean-reverting process.  And rather than being a static number, the mean is itself a process composed of long-term signal and short-term wave elements.  The final element of noise is determined by the distribution parameters including standard deviation, kurtosis and correlation—which are themselves mean-reverting processes—known as garch—also composed of signal and wave elements.  In summary, the intertemporal riskmodeling process deconstructs historical data into correlated signal, wave and noise—each of which is separately forecast—and then reconstructs within a Monte Carlo simulation environment in order to produce the forecasted portfolio distribution.

 

The Bernoulli Moment Vector

The Markowitz Model uses the mean to represent the forecast or reward and the standard deviation to represent the dispersion or risk—thus laying the groundwork for risk-reward efficiency analysis.  The basic method of moments is a simple procedure for estimating distribution parameters.  The mean is the first moment of a distribution and is calculated as the average value—and the standard deviation is the second moment and is calculated as the average deviation about the mean.  Intertemporal riskmodeling simply expands on this basic concept.  The Bernoulli Model also employs an expansion on the method of moments with the Bernoulli moment vector (ie. BMV) relating to the portfolio distribution.  The zero moment in the BMV represents exposure and is simply the intuitive concept of initial value exposed to change.  The fifth moment is VaL and represents a utilitarian translation of reward and thus an expanded definition of reward.  The sixth moment is VaR and represents the confidence level and thus an expanded definition of risk.

 

Conclusion

The word Renaissance means rebirth and was used to describe the era following the medieval period lasting from the fourth to the sixteenth century.  It was René Descartes (1596-1650) who broke the logjam by founding modern philosophy, modern mathematics and the Cartesian coordinates—all based on his belief that one should formulate a simple set of rules and follow them.  The method of moments represents a simple set of rules with the potential for advanced forecasting and risk-reward efficiency analysis.  Self-similarly, the BMV is the new Cartesian coordinates of the four-dimensional space-time continuum.  One might then pose the question—How long until the logjam breaks and the scientific management Renaissance emerges?

 


Scientific Management

Summary

Caption

Scientific Management

Essays

 


Summary

Scientific Management follows the development of relativity from Archimedes to Einstein—and then takes a parallel line of reasoning in considering the development of scientific management and portfolio theory.

 


Caption

Give me one fixedpoint and I will move the earth.

—Archimedes

 


Scientific Management

An Essay by Christopher Bek

 

Essay

Archimedes (287-212 bc) was the profoundly practical Greek genius of mathematics and physics who foreran many modern scientific discoveries.  He is ranked along with Gauss and Newton as one of the three greatest mathematicians of all time.  The calculation of a sphere’s volume and the realization of Archimedes’ principle—ie. Any object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced—count among his many greatest triumphs.  It could even be argued that his claim—Give me one fixedpoint and I will move the earth—metaphorically represents the starting point for all of science.  The need for a fixed starting point can also be found in ancient folklore—For want of a nail, the shoe was lost.  For want of a shoe, the horse was lost.  For want of a horse, the rider was lost.  For want of a rider, the battle was lost.  For want of a battle, the kingdom was lost.

 

Finding the Nail

The Greek Thales (624-546 bc) launched geometry as the very first mathematical discipline.  Thales in fact made his monumental contribution to mathematics after having amassed a personal fortune by first forecasting bumper olive crops and then purchasing options on the usage of olive presses.  The Greek Euclid (350-300 bc) carried on from Thales by developing Euclidean geometry as described by five simple axioms.  Euclid also made immeasurable contributions to mathematics with his landmark thirteen-volume masterpiece Elements.

 

Finding the Shoe

The French mathematician René Descartes (1596-1650) wrote the brilliant Discours de la méthode as a philosophical examination of the scientific method.  An appendix of the text presents the revolutionary idea of geometry as a form of algebra—ie. Cartesian coordinates.  After Descartes, the German Mozart-of-mathematics Carl Gauss (1777-1855) laid the foundation for non-Euclidean geometry by proving that additional geometrical systems exist in which only four of the five Euclidean axioms hold—and by arguing that there is no a priori reason for space not to be curved.

 

Finding the Horse

The Italian physicist and astronomer Galileo (1564-1642) laid down the fundamentals of the modern scientific method by developing a comprehensive, empirical approach to solving problems.  He was sentenced to life imprisonment at the age of sixty-nine for supporting the Copernican view that the earth revolves around the sun—a view which had the effect of destroying the notion of the earth as a fixedpoint.  Sir Isaac Newton (1643-1727) invented calculus, established the heterogeneity of light, and formulated the three laws of motion and the universal law of gravitation.  Both Galileo and Newton asserted, as far as relativity was concerned, that space itself was the universal frame of reference within which the freewheeling of stars and galaxies could occur.

 

Finding the Rider

In 1881, two Americans, Albert Michelson (1852-1931) and Edward Morley (1838-1923) performed a monumentally important experiment which established, beyond a doubt, that the speed of light is invariably fixed at 186,284 miles per second—regardless of relative motion.  In 1904 the Dutch physicist Hendrik Lorentz (1853-1928) formulated a group of algebraic transformations relating to electricity.  Mathematicians use transformations as tools for revealing fundamental underlying properties that remain invariant under transformation.

 

Winning the Battle

The Michelson-Morley experiment presented a problem in that, according to Newtonian physics, velocities are additive, thus contradicting the invariance of lightspeed.  The young Albert Einstein (1879-1955) resolved the dilemma in 1905 with his special theory of relativity by revealing that space and time are variable, interrelated quantities.  In paralleling Newtonian physics, Einstein theorized that the laws of nature are the same for all uniformly moving bodies.  But unlike Newtonian physics, which only concerns itself with mechanical laws, special relativity also accounts for the behavior of light and other electromagnetic radiation.  Einstein dismissed the separate notions of space and time by replacing them with the combined, fixed notion of spacetime.  In other words, according to special relativity, space and time are actually manifestations of each other.  Space becomes time and time becomes space at speeds approaching lightspeed in accordance with the transformations of Lorentz.

 

One Lazy Dog

At its inception, special relativity was little more than a set of algebraic equations that made only a modest impact.  It was not until 1909 when Hermann Minkowski (1864-1909) presented a geometric interpretation of relativity—as characterized by the four-dimensional spacetime continuum—that the scientific community took notice.  Ironically, Minkowski was also Einstein’s university professor and had described Einstein as a lazy dog who never bothered with mathematics at all—which makes sense given that Einstein sought elemental conceptual pictures first before considering mathematical complexities.  Einstein eventually warmed up to the idea of geometrization, and presented general relativity in 1915 as a geometric representation of special relativity that incorporates the concepts of gravity and curved space.  With special relativity, Einstein refurbished Newtonian physics in respect of uniformly moving bodies traveling along straight lines.  General relativity then upgrades special relativity so as to account for bodies traveling at varying speeds along curved lines

 

Losing the Kingdom in the Math

Einstein had become mathematically adept by the time the foundation of quantum theory was laid in 1925.  But unlike relativity, which concerns itself with the very large, quantum theory is directed towards understanding the very small.  In fact, the great revelation of quantum theory is that atomic phenomena are indeterminate—thus making it a statement of probability.  But Einstein did not believe in cosmic risk, as evidenced by his claim that God does not play dice.  And so in dismissing the notion of risk, Einstein actually overlooked the makings of a cosmic fixedpoint.  He eventually strayed from his original conceptual approach and spent much of the last thirty years of his life withdrawn in the world of obscure mathematics and twisted geometries.

 

Capitalism’s Fixedpoint

Harry Markowitz developed portfolio theory as a way of constructing optimally balanced portfolios that maximize reward for given levels of risk.  Markowitz forever linked reward with risk in the same way that Einstein linked space with time in that both the expected outcome and the attendant uncertainty of outcome are required to complete the picture.  Markowitz wrote a fourteen-page paper entitled Portfolio Selection as a graduate student at the University of Chicago in 1952—and then shared the Nobel Prize for economics with two others in 1990.  His original approach employs simple matrix algebra to aggregate risk—and applies linear programming algorithms to then determine optimal asset allocations.  In many ways the discoveries of Einstein and Markowitz parallel each other.  Both men were unknown mid-twenty year-olds who wrote brief, modest papers with far reaching implications.  The main difference so far being that we have yet to realize the potential for portfolio theory.

 

Kill all the Accountants

While Galileo and Newton believed space itself to be the fixedpoint of the cosmos, most companies today subscribe notionally to the market as capitalism’s fixedpoint.  Yet a closer look at the market reveals it to be mostly neurotic and chaotic—and anything but fixed.  And so if companies genuinely wish to acquire a fixed, internal decisionmaking frame of reference, they must resolve to undertake the scientific work necessary to properly implement portfolio theory.  While Einstein lost the holy grail of physics, unified field theory, in the math—companies today are losing the holy grail of capitalism, shareholder value, in the balance sheet.  The great challenge now lies in learning to transform the entrenched accounting mindset into the new scientific management mindset.

 

Conclusion

Consider how the notion of connecting two personal computers with modems a few decades ago has managed to become the all powerful internet.  Then consider how corporate officers and directors can begin rolling out scientific management by first nailing down scientific management conceptually in their minds—never forgetting that science is not a spectator sport.  Scientific management is firstly an exercise in seeing the forest for the trees—and secondly an exercise in seeing the trees for the forest.  Portfolio theory gives us the overall perspective of the forest, while scientific management tools like Monte Carlo simulation, intertemporal riskmodeling and advanced optimization algorithms have the potential, in the hands of scientists, to provide us with an accurate representation of the trees.

 


Table of Contents

Introduction

Christopher Bek Résumé

The Bernoulli Model

Risk Management Review Essays

Quotations