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 decision making lead to the efficient frontier.
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 Geometry. 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.