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. There is no more commonplace statement than the world in which we live is a four-dimensional spacetime continuum. —Albert Einstein
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 in 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. Galaxy clusters, earthquakes, mountains, snowflakes, lightning, and broccoli are just a few of the naturally occurring phenomena exhibiting fractal qualities. The power of fractal analysis lies in the ability to capitalizes on self-similarity across scale by locating an eerie kind of order lurking beneath seemingly chaotic surfaces. Owing to its mathematical basis, fractal analysis is scalable across and between applications.
Lost Moments in Time. While the coastline conundrum involves fractal analysis for measuring the complexity of geometrical shapes, the British hydrologist Harold Hurst (1900-78) employed fractal risk analysis to manage the Nile river dam from 1925 to 1950—with the goal being the balancing of overflow risk against the risk of insufficient reserves—a job description not unlike that of a treasurer. 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 of spacetime such that the laws of physics no longer apply. Examples of singularities include the big bang, black holes and one divided by zero. What physicists like Stephen Hawking who developed the concept failed to realize is that a breakdown of 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 fractal or stable Paretian—named after Vilfredo Pareto (1848-1923). The three explicit fractal distributions are the Bernoulli (ie. coin toss), normal and Cauchy—relating to fractality 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. Depending on the fractality, the Camus interpolates between the Bernoulli and the normal or the normal and the Cauchy. 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 a greater potential for larger jumps.
Intertemporal Riskmodeling. The Hurst Model 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 itself is a process composed of long-term signal and short-term wave elements. The final element of noise is determined by the distribution and correlation parameters—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 term Renaissance means rebirth and described 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 the belief that one should formulate a simple set of rules and follow them. The method of moments represents a simple set of rules for advanced forecasting and efficiency analysis. Self-similarly, the BMV represents 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?