Summary—The Metaphysics of Risk tells the story of fractals (ie. fractions of dimensions) in the modeling of risk.
Erwin Schrödinger (1887–1961) was an Austrian physicist and Nobel laureate best known for his study of the wave mechanics of orbiting electrons used in formulating quantum theory. In attempting to explain the paradoxical nature of quantum mechanics, Schrödinger put forth the classic cat-in-a-box thought problem as follows—A quantum-cat is placed in a box. The box is such that no one can know what is happening inside. A device triggers the release of either food or poison with equal probability. And the cat meets its fate—or does it? In the strange world of quantum mechanics, subatomic particles exist in several places at once, and only become determinate upon observation. Schrödinger argued that the cat is both alive and dead until the moment the box is opened. Inside the box, unobserved, the state of the cat exists only as a probability wave. Quantum theory essentially says that beyond a certain point, known as Planck’s constant, the universe is indeterminate—thus quantum theory is effectively nothing more than an attempt to understand the essence of risk.
Metaphysics. Metaphysics is a branch of philosophy concerned with the ultimate nature of reality. Ontology is a further branch of metaphysics that serves to catalogue the fundamental, distinct elements or dimensions that constitute reality. The physical world is made up of space, time and matter, whereas ontological dimensions of the metaphysical world include consciousness, self-awareness and God. In describing the universe and the relativity of space and time, Einstein said shortly before the discovery of quantum theory that—There is no more commonplace statement than the world in which we live is a four-dimensional space-time continuum. Among other things, his claim is that reality is composed of more things than meet the eye. One might even argue that Einstein underestimated the dimensionality of the universe. Consider the possibility that reality is not just the four-dimensional place that Einstein suggested, but is in fact an n-dimensional continuum composed of all ontological dimensions—both physical and metaphysical. And since quantum theory clearly proved that risk is indigenous to the universe—it must also be considered part of the continuum.
Fractional Dimensions. In 1975 the Polish mathematician Benoit Mandelbrot, posed the question—How long is the coastline of Britain? Appealing to relativity, Mandelbrot pointed out that the answer depends on one’s perspective. From space, the coastline is shorter than it is to someone who is walking. That is because on foot the observer is exposed to greater detail and must travel farther. Ultimately, according to Mandelbrot, when the shape of each pebble is taken into account, the coastline turns out to have infinite length. Mandelbrot proposed a system for measuring irregular shapes by moving beyond integer dimensions to the seemingly absurd world of fractional dimensions. He used a simple procedure involving the counting of circles to calculate what he called 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.
The Nature of Fractals. Fractals are a way of measuring phenomena that are otherwise immeasurable. In the perfect world of Platonic harmony, fractals would have no meaning. However, in the real world, fractals give the universe its form. A fractal is a geometrical shape having the property of self-similarity in that each small portion can be viewed as a reduced scale replica of the whole. Clouds are fractal as evidenced by the fact that they look the same up close as they do from a distance. Mountains, broccoli, lightning, galaxy clusters, earthquakes and snowflakes are a few of the naturally occurring phenomena exhibiting fractal qualities. Fractals are characterized by local randomness and global determinism. Determinism provides natural order, while randomness gives us innovation and diversity. One pine tree looks much like the next. And branches are similar to one another in structure, yet increasingly random as they travel away from the tree trunk. By capitalizing on the strength of self-similarity across scale, fractals are able to unmask the eerie order lurking beneath chaotic surfaces. Fractals take systems of unfathomable complexity when viewed in context of conventional geometry—and transforms them into structures of lucid simplicity.
Fractal-Powered Evolution. The evolutionary development of life has exhibited consistent patterns across biological scale. As well, the timing of evolution is fractal in that life has advanced in clusters followed by long periods of stagnation. Computerized genetic algorithms imitate evolutionary fractal patterns as a way of capturing the amazing capacity of life. The constant shifts in one’s attention between mindless-trivia and timeless-verities follows a fractal pattern that reflects the fractured, yet coherent, structure of consciousness. This structure resonates throughout all levels of one’s persona. Artificial neural networks are designed to mimic the mind in an attempt to harness the fractal power of consciousness. Geologists and geophysicists apply fractals in modeling drainage networks, erosion, floods, earthquakes, petroleum reservoirs, mantle convection and magnetic field generation. Tire manufacturers use fractals to design patterns that optimize road adhesion. Fractals are also used in algorithms for compressing computerized graphics files.
Fractal Time. Fractals exist in nature because theirs is the most stable and error tolerant structures. Consider a typical financial market made up of a large number of investors—each with different time horizons. All investors face the same risk exposure—given the respective time horizon. For example, both an investor with a one-day horizon and an investor with a one-year horizon are equally likely to face crashes during their respective time horizons. Yet a crash for the day-investor may be seen as a buying opportunity for the year-investor. The market remains stable because it has no inherent time horizon and therefore, in this sense, is fractal.
Fractal Risk. According to the efficient market hypothesis, a market is efficient because it consists of a large number of rational, well-informed, profit-seeking, risk-averting investors. The theory holds that investors act on information as it becomes available and, as such, the markets quickly incorporate new information. In reality, investors tend not to act on information until it becomes an established trend. Then, they do so more with a herd-mentality rather than as rational freethinkers. The efficient market hypothesis leads to the conclusion of a market exhibiting normally distributed price changes. Yet it follows from the discussion above that the distribution of changes should display fatter-tails and a higher-peak than those inherent to the normal distribution. And in reality, that is exactly what happens. The fatter-tails occur in the distribution of changes because there are more big-jumps than one would normally expect. Similarly, a higher peak is the result of a greater frequency of periods with little of no change in rate.
The Camus Distribution. Mandelbrot put forth the fractal distribution, which embodies fractal characteristics like self-similarity. It allows for fatter-tails and a higher-peak than those of the normal distribution. In fact, the fractal distribution is a superset of the two-moment (ie. mean, standard deviation) normal distribution which when combined with the infinite-moment Cauchy distribution produces the four-moment (ie. mean, standard deviation, skewness and kurtosis) Camus distribution that I developed. The normal distribution is related to the Cauchy distribution as demonstrated in the Microsoft Excel simulation—ie. normal =normsinv(rand()) and Cauchy =normal1/normal2. The Camus distribution includes two extra parameters (ie. skewness and kurtosis), which are calculated as a function of the fractal dimension. When the fractal dimension is a whole number, the fractal distribution reverts to the normal distribution. In other words, the assumption of normality only holds true in regard to unfractured phenomena—which do not exist in the real world.
Portfolio Selection. In 1952 Harry Markowitz put forth a ground-breaking, fourteen-page paper entitled Portfolio Selection. The essence of the theory involves the concatenation of three algorithms that are forecasting, integration and optimization. The entry-level algorithms used in portfolio theory are regression analysis, the central limit theorem and linear programming. The Bernoulli Model that I developed is an advanced application of portfolio theory that presents the same consistent storyboard for all organizational risk factors. The storyboard sits atop a stylishly engineered portfolio of scientific management algorithms that form an advanced forecasting system that is mathematically accessible to executives. The Bernoulli Model uses a breakdown of time-series data into signal, wave noise. Advanced-level algorithms used in portfolio theory include garch analysis (ie. advanced regression analysis), Monte Carlo simulation with the Camus distribution—and hill-climbing and genetic optimization algorithms.
Conclusion. As an application of philosophy, nowhere is it more important for the rubber to firmly meet the road than with respect to understanding risk. Fractal analysis captures a whole other dimension of complexity beyond that of traditional approaches. Alternatively, in considering where the rubber meets the sky, one may even wish to reflect on one’s own fractal nature.