# The Bernoulli Model

## Caption

Ever since I was engaged in writing Principia Mathematica I have come to realize a certain method which consists of working towards building a bridge between the commonsense world and the world of science.  I accept both, in broad outline, as not to be questioned.  But as in cutting a tunnel through mountain, work must proceed from both ends in the hope at last that the painstaking labour will be crowned with a meeting somewhere in the middle.

—Bertrand Russell

## The Bernoulli Overview

#### The Bernoulli Abstract

The Bernoulli Model is a risk management and decisionmaking methodology 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 which is mathematically accessible to executives.  It is named after a family of Swiss mathematicians who lives several hundred years ago and is founded on portfolio theory developed by Harry Markowitz at the University of Chicago in 1952.  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.  The Markowitz Model concatenates three algorithms—forecasting, integration and optimization—in constructing a holistic portfolio optimization algorithm that serves to maximize reward for given levels of risk.  Building on Markowitz, The Bernoulli Model expands along a multitude of dimensions including forecasting, efficiency analysis, utilization, accountability and comparability.  The Bernoulli Model further adds the Delphi process to more specifically define organizational values—utility theory to translate external values into internal values—Monte Carlo simulation to aggregate heterogeneous risk components—the Camus distribution to four-dimensionally represent risk—The Bernoulli Moment Vector (BMV) for tracking forecasts—and an alternative hypothesis to serve as the loyal opposition to the null hypothesis—and highly advanced Excel charts.

#### The Bernoulli Concept

The Bernoulli Model represents a fundamentally different methodology to managing risk and value creation.  The essence of the difference is found in three salient points.  Firstly, it involves a shift from an external, market-based focus to an internal, mathematically-based focus.  Secondly, The Bernoulli Model is a top-down approach that is in stark contrast to existing bottom-up approaches.  The Bernoulli Model is not intended to replace existing systems but to give executives a thoroughly different perspective.  Thirdly, it is based in the common software applications of Microsoft Excel, Visual Basic and Access and includes all related material so that organizations may, if they choose, carryon developing the model on their own without reliance on outside consultants and software updates.  The use of Microsoft Access as the database for The Bernoulli Model means that it can easily be migrated to large-scale database systems.  The Bernoulli concept involves building a foundation by developing micromodels for the purpose of highlighting basic subjects.  Potential micromodel subjects for The Bernoulli Model include market risk, credit risk, the forward curve, drilling risk, capital investment risk, intertemporal riskmodeling, insurable risk, Bayesian analysis and expert opinion, game theory and the Delphi process.  The Bernoulli concept includes all related material such as the Excel charts, Visual Basic code, Access databases, RoboHelp files, as well as the Excel models themselves.

#### The Bernoulli Archaeology

In 1670 both Newton and Leibnitz formulated versions of calculus—the mathematics of motion.  John and James Bernoulli picked up on calculus, spread it through much of Europe, and set the roadmap for efficiency analysis by finding the curve for which a bead could slide down in the shortest time.  John’s son Daniel set the roadmap for utility theory based on the idea that the value of an asset is determined by the utility it yields rather than its market price.  No less than eight Bernoulli’s made significant contributions to mathematics.  In 1952 a twenty-five year-old University of Chicago graduate student named Harry Markowitz stood on the shoulders of giants like Archimedes, Descartes, Newton, Leibnitz, Pascal, Bayes, Cauchy, Gauss, Galois, Laplace, von Newmann, Einstein and the Bernoulli’s in producing a fourteen-page paper entitled Portfolio Selection.  His approach combines regression analysis with matrix algebra and linear programming in the engineering of basic portfolio theory.  The Bernoulli Model builds on Markowitz by adding components like the Delphi and expert opinion programs along with utility theory and event riskmodeling using decision trees—and by fortifying existing components with advanced regression models and metaheuristic algorithms such as Monte Carlo simulation, neural networks, genetic and hill-climbing algorithms, and the state-of-the-art four-moment Camus distribution.  The Bernoulli Model is a stylish Excel-based, RoboHelp-complemented, totally-expandable, enterprise-wide, actuarial-valuation, decision-making system.

#### The Bernoulli Architecture Delphi Program The Delphi program allows for the input of organizational values which then guides the decisionmaking process.  The basic Delphi value is represented by a confidence level describing the maximum allowable downside change in portfolio value—ie. VaR.  Advanced Delphi values include utility translations and objectives relating to financial, strategic, operating and competition. Forecasting The process of forecasting produces not only estimates of outcomes but also estimates of uncertainty surrounding outcomes.  Advanced forecasting methods include forward curve analysis, option price analysis, intertemporal riskmodeling, expert opinion, Bayesian analysis, neural networks and event riskmodeling. Integration In addition to the basic closed-form method of integration involving the two-moment normal distribution, the Bernoulli Model also employs Monte Carlo simulation along with the four-moment the Camus distribution in order to capture and integrate the full spectrum of heterogeneously distributed forecasts of outcomes and uncertainty surrounding outcomes. Optimization Optimization algorithms search risk-reward space in order to determine the optimal set of decision subject to Delphi constraints.  Closed-form optimization algorithms include linear programming while open-form methods include hill-climbing and genetic algorithms. Optimized Portfolio The optimized portfolio represents the raison d’être portfolio theory.  Portfolio theory brings together the consequences of a varied set of uncertain components.  The Bernoulli Model employs sensitivity analysis or stress-testing algorithms to insure optimal portfolio robustness. Historical Data Historical data includes market rates, forward rates, option rates and production data—both historical and current.  In addition, portfolio accountability analysis is fed back into the model along with the other historical data in order to essentially make the forecasting process self-aware. Exposure Data Exposure is simple the initial asset value exposed to change.  For example, five barrels of oil at \$40 per barrel equals \$200 of exposure.  A long position is has positive exposure while a short position is negative.  Exposure data also includes exposure dynamics particularly relevant to risk components like credit risk. Expert Opinion Expert opinion is particularly useful when historical data is unavailable or unreliable.  The Bernoulli Model integrates expert opinion into the forecasting process that uses Bayesian analysis to estimate uncertainty surrounding forecasts.  The model also provides experts with regular follow-up performance reports. Event Scenarios One of the first event risk studies was conducted in 1933 to examine the effects of stock splitting on price.  Event riskmodeling is an advanced form of forecasting that uses decision trees in order to see how specific scenarios might play out.  The approach is well suited to contingency planning. Comparability Analysis While the primary function of portfolio theory is to bring together all uncertain components into a single view, the secondary function is to provide comparability between components.  The Bernoulli Model employs the complimentary principle with the null and alternative hypotheses. Accountability Analysis The nineteenth century saw the inauguration of management accounting designed to manage the efficient conversion of raw materials into finished products.  The Bernoulli Model is designed to manage the efficient conversion of uncertain information into organizational value.

#### The Bernoulli Features

·         Roadmaps covering all risk factors

·         Top-down implementation potential

·         From training-level to fully-operational

·         Expanded definitions of both risk and reward

·         The realization of organizational portfolio management

·         An enterprise-wide, actuarial-valuation, decision-making system

·         The state-of-the-art four-moment Camus distribution for modeling risk

·         The Bernoulli Moment Vector (BMV) for tracking asset values and forecasts

·         An engineered help system to perfectly complete the Bernoulli theatre of online scientific management

#### The Bernoulli Paradigm

·         Building the case

·         The complementary principle

·         From accounting to scientific

·         From a posteriori to a priori

·         Predefined values and opinions

·         The frontier of asset management

·         The awesome power of prototyping

·         Do you know what a paradigm shift is?

## The Bernoulli Slides

### The Bernoulli Delphi

#### Delphi Worksheet The Delphi definition worksheet is the overriding guidance system of The Bernoulli Model—which 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 method 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—also known as value-at-risk or VaR.  The secondary Delphi value pertains to the utility translation function, which adjusts market returns to more accurately represent internal organizational values.  If the Delphi process were applied to a sovereignty, then the outcome would be a constitution.  The political theorist Jean Jacques Rousseau (1712-1778) rejected the notion of representative democracy and instead asserted that everyone should vote on every issue.  The Delphi offers a more enlightened solution with the predefinition of values.  The Delphi definition worksheet also includes two additional charts—The Fractal Scaling Definition, which relates the kurtosis to the fractal scaling exponent—and The Bernoulli Form, which depicts the touchstone for The Bernoulli Model by graphically representing the solution to the problem of the curve for which a bead can slide down in the shortest time.  Below is a list of possible additional Delphi criteria.

 Financial Objectives Cash flow, dividends, earnings, interest coverage, shareprice, debt rating and financial solvency concerns. Strategic Objectives Basic organizational objectives. Operating Objectives Protection from property, business interruption and liability losses. Competitive Objectives Objectives regarding positions relative to competitors. Directors and Officers Bias Liability protection for directors and officers of the organization. Social Values Needs of the employees and the community.

#### Delphi Parameters Valuation Parameters Portfolio Increment Portfolio increment describes the size the portfolio under consideration. Epoch Units An epoch is a basic unit of time.  The basic unit of time for The Bernoulli Model in one month. Valuation Epochs In this example the Delphi process has determined the global valuation period is three months. VaR Definition Frac—M9 The fractal scaling switch determines whether to adjust risk normally (ie. with the square root of time—0.5) or fractally (ie. with an exponent between 0.5 and 1.0 in accordance with the transformation of kurtosis. DelphiVaR The global DelphiVaR determines the global value (ie. VaR—value at risk) for downside risk exposure. DelphiCL The global DelphiCL determines the global confidence level (ie. CL) for downside risk exposure. DelphiSD The global DelphiSD is the calculated number of standard deviations (ie. SD) based on the CL and the Frac—M9 switch. Utility Definition The utility definition shows the unadjusted and adjusted returns at three points and the parameters that create the curve which translates from unadjusted to adjusted returns. Workbook Settings Window Zoom The window zoom is a convenience feature that allows users to zoom all worksheets at once. Smart Rounding The smart rounding is a display feature which formats display into the specified number of significant digits.

#### Utility Definition The utility definition converts unadjusted returns into adjusted returns in accordance with the utility transformation curve.  So while the four-moment Camus distribution and fractal scaling provides an expanded definition of risk, utility theory provides an expanded definition of reward by changing external market values into internal Delphi values.  For example, a 100 percent return (ie. Mu—M1) becomes a 50 percent return (ie. VaL—M5) while a –20 percent return (ie. Mu—M1) becomes a –30 percent return (ie. VaL—M5).  The rationale being a reflection of the gravity of the return.  A 20 percent loss in value could well have unexpected secondary effects that could end up feeling like a 30 percent loss.  Similarly, while a 100 percent return is obviously a desirable outcome, undue emphasis on trying to achieve such a result may produce erratic outcomes and the missing out on more conservative opportunities.  Therefore, under utility translation, the 100 percent return is viewed internally as just a 50 percent return.  Utility theory was founded by Daniel Bernoulli (1700-82) with his 1738 paper entitled Exposition of a New Theory on the Measurement of Risk—with its central theme being that the value of an asset is the utility it yields rather than its market price.  His paper, one of the most profound documents ever written, delineates the all-pervasive relationship between empirical measurement and gut feel.

#### Fractal Definition The fractal definition determines the scaling exponent (ie. Frac—M9) as a function of kurtosis (ie. Kurt—M4).  A normal distribution has a kurtosis of three which translates into fractal scaling exponent of 0.5—which is the normal application with the square root of time.  Going from a one-month valuation period to a one-year valuation period under a normal assumption (ie. M4 = 3, M9 = 0.5) results in a scaling factor of 3.5 times (ie. 12^0.5)—while a similar calculation under a Camus distribution with a kurtosis of six (ie. M4 = 6, M9 = 0.75) produces a scaling factor of 6.4 times (ie. 12^0.75).  The rational is simply that with higher kurtosis comes the potential for larger jumps and thus a larger scaling factor.  The fractal scaling exponent is also known as the Hurst exponent named after the hydrologist Harold Hurst (1900-78) who managed the Nile River Dam from 1925 to 1952 by formulating water discharge policies aimed at minimizing both overflow risk and the risk associated with insufficient water reserves.  Fractals are self-similar geometric shapes in that there is no inherent scale so that each small portion can be viewed as a scaled replica of the whole.  Fractal analysis in riskmodeling capitalizes on self-similarity being able to extrapolating small-scale data (eg. daily) to larger-scale data (eg. monthly) where unavailable.  Fractals exist pervasively in nature because theirs is the most stable and error tolerant.

#### The Bernoulli Form The Bernoulli Form is the metaphor of efficiency that undergirds the entire Bernoulli methodology.  The Bernoulli Model is mathematically and ascetically efficient in its singular direction towards organizational efficiency.  Efficiency is the ability to achieve an objective with a minimal expenditure of resources.  In the case of risk-reward efficiency, the objective is to achieve the highest expected reward with the least expected exposure to risk.  In the case of cost-function efficiency, the objective is to achieve optimal function with the least expenditure of cost.  The Bernoulli brothers, James (1654-1705) and John (1667-1748), challenged each other with the problem of finding the curve for which a bead could be slide down in the shortest time—and found the answer in a cycloid—thereby laying the foundation for efficiency analysis.  John’s son Daniel set the roadmap for utility theory and also formulated the Bernoulli principle which simple state that the faster gas flows in a pipeline the lower the pressure on the walls of the pipe.  Airplane wings are designed in accordance with the Bernoulli principle so that the distance air travels over the top of the wing is greater than the distance it travels under the wing thereby creating a pressure differential and thus lift.  No less than eight Bernoulli’s made significant contributions to mathematics.

### The Bernoulli Exposure Exposure is simply the initial asset value exposed to change.  For example, five barrels of oil at \$40 per barrel equals \$200 of exposure volume.  A long position has positive exposure while a short position has negative exposure.  Exposure data also includes exposure dynamics particularly relevant to risk factors like credit risk.  Exposure status allows users to turn specific exposures on and off.  Note that the sensitivity switch in the valuation parameters allows for inclusion of negative and positive exposures.  Contract RateM0 is the initial contracted rate of the position and also represents the initial exposure for analysis of the rate.

### The Bernoulli Analysis

#### Technical Analysis Worksheet If the Delphi definition worksheet is the overriding guidance system of The Bernoulli Model, then the technical analysis worksheet is the engine, the actuarial valuation worksheet is the cockpit and the prechart analysis worksheet is the transmission.  The Bernoulli technical analysis worksheet contains a systematic delineation of the technical analysis involved in the actuarial valuation process.  The Bernoulli Model is a prototyping system where the micromodels serve as both templates and checkers for the operational models.  The micromodels, combined with the Bernoulli on-line help system, may also act as training models for executives and managers.  Ultimately it means that every asset management process in the enterprise-wide actuarial valuation system is accessible to executives.  The Harvard Business Review publication began in 1922 with the mandate of connecting fundamental economic and business theory with executive experience.  The Bernoulli Model represents a systematic realization of that very mandate.  It is designed for everyday use by executives, managers and analysts.  The model offers the ability to either skim the surface of riskmodeling and decision analysis or, alternatively, to drill-down and examine the inner workings of specific decision problems.  Ultimately it frees up executives from the politics of decisionmaking and allows them to think more broadly about values and objectives. ### Prechart Analysis Worksheet The Bernoulli prechart analysis worksheet contains the translation of the technical analysis into the graphical format found in the six charts on the actuarial valuation worksheet.  The charts themselves are very sophisticated and use well-written Visual Basic code to generate the chart data—all of which is available for use in other applications and as the starting-point for other charting applications.  The ability to present ideas in a consistent and uncluttered way is the hallmark of The Bernoulli Model.  The concept of a theory comes from the ancient Greek idea of detachment as the path to wisdom.  The word theory is the root of the word theatre and is derived from the Greek verb theatai—which means to behold as an action in which the observer is not involved.  In fact, the dirty little secret of all great thinkers is that new ideas are formulated completely outside any everyday view of reality.  Einstein first introduced relativity theory in 1905 as a simple set of algebraic equations.  Yet the theory was largely ignored until four years later when Minkowski presented a geometric view of relativity as characterized by the four-dimensional spacetime continuum.  The Bernoulli Model represents a riskmodeling and decisionmaking theatre for executives, managers and analysts alike.

### The Bernoulli Valuation

#### Actuarial Valuation Worksheet The Bernoulli Model presents the same consistent storyboard for all organizational risk factors.  The display parameters are on the top while the valuation parameters are on the left.  Below the valuation parameters is the component Bernoulli Moment Vector (BMV) while on the right side of the charts is the portfolio BMV.  If the three charts on the left of the storyboard are the components or ingredients in a loaf of portfolio bread, the three charts on the right are the different ways of slicing up the bread.  The light blue is the null paradigm—green is the alternative paradigm—dark blue is the common between the two paradigms.  Chart V1 delineates the risk factor exposure to change in value—eg. five barrels of oil at \$40 per barrel equals \$200 of exposure.  Chart V2 captures the component forecast distributions for two of the components.  Chart V3 shows the correlation between changes in value of components—ie. a correlation of one means factors move in lockstep and a correlation of zero means no correlation.  Chart V4 illustrates first-order risk management by contrasting risk (ie. VaR) with risk exposure limits (ie. Delphi).  Chart V5 captures the portfolio forecast distributions and demonstrates the standardized Bernoulli paradigm frame—ie. –/+ six-sigma.  Chart V6 illustrates second-order risk management by contrasting value creation against the risk associated with value creation.

#### Display Parameters HistoIter HistoIter determines the number of histogram iterations or statistical samples used to generate the distributions in Charts V2 and V5. V23Disp The V23Disp switch allows for toggling between rate and portfolio in Charts V2 and V3.  The rate setting shows value before exposure, while the portfolio setting shows value after exposure. BMVdisp The BMVdisp changes the display base of the Bernoulli moment vector (ie. BMV).  Value shows asset values.  M0 shows percentage of exposure.  M2 shows the number of standard deviations. V25Axis The V25Axis switch changes the X-Axis for Charts V2 and V5.  Value shows asset values.  M0 shows percentage of exposure.  M2 shows the standard six-sigma paradigm frame. TailScale The TailScale switch works in conjunction with the TailMult switch and allows for settings of Off, 3SD and VaR.  If only one ValStat switch is on and if the TailScale switch is on, then the tails are dark blue. TailMult The TailMult switch works in conjunction with the TailScale switch—and is the tail multiplication factor.  The valid range for TailMult is zero or greater.

#### Valuation Parameters ValStat The ValStat switch turns the Null and Alt valuation paradigms on and off.  The light blue is the null paradigm—the green is the alternative paradigm—the dark blue is the common between the two paradigms. ValForm ValForm determines whether the actuarial valuation is closed or simulated.  Closed-form valuation utilizes matrix algebra and the normal distribution to integrate forecasts, while the simulated-form is much more robust and is able to make use the four-moment Camus distribution and the utility translation feature. ValDate ValDate is the date at which the assets are valued. ValType An ex ante valuation uses only historical data, while an ex post valuation uses both historical and future data—ie. perfect information.  The overriding goal of forecasting is to construct models that best align ex ante with ex post results. Epochs Epochs refers to the number of non-zero weighted epochs. Lambda Lambda determines the decay weights between adjacent epochs. Sensitvty The Sensitvty parameter is related to exposure and allows for either short positions, long positions or both. Utility The Utility switch determines whether the utility transformation defined on the Delphi definition page is activated or not.  The Utility transformation only works if the ValForm switch is set to Sim. M1stat In the on position M1stat uses the forecasted value, while in the off position the expected value of the forecast is zero. MMstat The MMstat switch has three settings—off, M22, M44.  The off setting produces expected correlations of zero, while the M22 setting produces the forecasted correlations.  The M44 setting stress-tests correlations. SimIter SimIter determines the number of simulation iterations used to calculate the BMV when the ValForm switch is set to Sim. Dist The Dist switch selects the distribution as either the normal or the Camus.  The normal is a two-moment distribution while the Camus is a four-moment distribution. PortM3 The PortM3 input allows for the third-moment or skewness of the portfolio to be overridden only if the Dist switch is set to Camus. PortM4 The PortM4 input allows for the fourth-moment or kurtosis of the portfolio to be overridden only if the Dist switch is set to Camus. Offset The Offset switch is a display switch which determines the offsets to be overridden—with the choices being MTV or return.  MTV is mark-to-value and is the change since the .  Return . Marker The Marker switch toggles the triangle indicating M1, M5 and Sim for Charts v2 and v5.

#### Component Exposure Chart Exposure is simply the intuitive concept of the effective asset value exposed to change.  For example, five barrels of oil at \$20 per barrel equals \$100 of exposure.  For comparison, a penalty in hockey represents two-minutes of powerplay exposure.  A long position has positive exposure while a short position is negative.  Exposure is essentially the denominator of risk analysis and includes adjustments for exposure dynamics and nonlinearities.  The model also calculates the effective portfolio exposure so that the notion of exposure is defined everywhere that risk is defined.  The Bernoulli Model uses exposure as the starting point for the actuarial valuation process and allows for full comparability between risk components in respect to both exposure and distributions.  The light blue represents the null paradigm.  The green represents the alternative paradigm and the dark blue is the common between the two paradigms.  The first four moments of a statistical distribution are the mean, standard deviation, skewness and kurtosis.  The mean is calculated as the average value, the standard deviation is calculated as the average deviation about the mean, the skewness is calculated as the average cubed deviation about the mean, and kurtosis is calculated as the average deviation to the fourth power about the mean.  In The Bernoulli Model, exposure is the zero or null moment.

#### Component Distribution Chart While exposure represents the denominator of risk analysis, the statistical distribution represents the numerator.  And while Chart V2 shows components distributions, Chart V5 shows the portfolio distributions for both the null and alternative hypotheses.  Both Chart V2 and V5 use the standard Bernoulli paradigm frame (ie. +/– six-standard deviations) and are affected by the V23Disp, V25Axis, HistoIter, TailScale and TailMult display switches.  Chart V2 can show either the rate (shown above) or the portfolio (shown below) distributions.  The rate distribution is prior to the application of exposure while the portfolio distribution shows the selected portfolio component after the application of exposure.  While the two charts shown here only show value along the bottom-axis, the V25Axis switch allows for both exposure and standard deviation along the bottom-axis.  In fact, the alternative paradigm (ie. the green) is calibrated to the standard Bernoulli paradigm frame (ie. +/– six-standard deviations).  The two available valuation methods are closed (ie. matrix algebra) and open (ie. Monte Carlo simulation).  Using Monte Carlo simulation and the four-moment Camus distribution, the Bernoulli Model is able to use the method of moments to capture the first four moments from the simulation and then apply it to the Camus distribution with the end result being a smoothly presented statistical distribution generated with a minimal number of simulation iterations. #### Component Correlation Chart While volatility of components is usually considered to make up the principal composition of a portfolio distribution, it is often dramatic changes in correlation between components in times of turbulence that leads to unforeseen shifts in portfolio value.  Chart V3 shows the expected correlation between changes in value of risk components.  A correlation of one means that the components move in lockstep, while a value of zero indicates there is no correlation between components.  While standard deviation is the second moment of a risk component, correlation is the second moment between risk components.  To more accurately capture kurtotic correlation (ie. large swings) the MMstat switch in the valuation parameters includes an M22 option (ie. normal correlation) and an M44 option, which essentially selects the correlation against the portfolio and in effect represents a stress-test of correlation.  Niels Bohr (1885-1962), one of the founding fathers of quantum theory, defined the complementary principle as the coexistence of two necessary and seemingly incompatible descriptions of the same phenomenon.  One of its first realizations dates back to 1637 when Descartes revealed that algebra and geometry are the same thing—analytic geometry.  The ability to contrast paradigms, scenarios, strategies, risk components and valuation parameters—and thus provide complementary perspectives of the same portfolio—represents an invaluable feature of The Bernoulli Model.

#### Portfolio VaR/Delphi Chart Value-at-risk or VaR is a market risk measurement standard that was created to allow companies to relate the income from trading to the risk inherent in trading.  Its original purpose was to establish a frame of reference for evaluating trades, as well as providing a mechanism for officers and directors to set risk exposure limits.  Although VaR is nothing more than a subset of The Markowitz Model formulated in 1952, the modern VaR movement started with the investment bank JP Morgan in its attempt to create a market risk measurement standard.  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 is that VaR is an unsophisticated measure and traders are now gaming VaR.  VaR is defined as a statement of probability regarding the potential change in value of a portfolio resulting from changes in market factors over a specified time interval, with a certain level of confidence.  A one-day time horizon and a confidence level of 95-percent are commonly used.  The Bernoulli Model uses a one-month time horizon as a basis and forecasts as far forward as twenty years.  The method included all organizational risk components and uses highly advanced forecasting techniques like intertemporal riskmodeling and decision trees with Monte Carlo simulation.  The Delphi process is used to establish the VaR confidence level and value.  Chart V4 shows the VaR/Delphi comparisons for the portfolio and three components.

#### Portfolio Distribution Chart 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 surrounding outcomes.  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.  There are explicit expressions for three fractal distributions—the Bernoulli (ie. coin toss), the normal and the Cauchy.  The Bernoulli converges to the normal distribution when the number of coins becomes sufficiently large.  The Cauchy is interesting in that it possesses undefined moments.  The Bernoulli Model uses the four-moment Camus distribution to model the full range of fractal distributions.  The first four moments of a statistical distribution are the mean, standard deviation, skewness and kurtosis.  The Bernoulli Model is able to capture the output from the most complicated forecasting simulation and present the smoothly-represented Camus distribution generated with a minimal number of simulation iterations.  The three images of Chart V5 here show the portfolio distributions for both the null and alternative hypotheses, for just the null, and for just the alternative—with a base of asset values, percentage of exposure, and the standard Bernoulli paradigm frame (ie. +/– six-standard deviations).  Note that the display is not limited to the specific combination of formats shown here.  #### Portfolio Efficiency Analysis Chart The efficient frontier has come to form the bedrock of modern portfolio theory since its introduction by Harry Markowitz in 1952.  It is the asset allocation mechanism of choice for virtually all pension funds, and pension fund money makes up the lion’s share of investments on Bay and Wall streets.  The efficient frontier is defined as the risk-return tradeoff curve.  The principle of risk and return is fundamental to asset management.  They are complementary items in that a greater return on investment is expected as more risk is accepted.  Markowitz defined risk as the variability of returns as measured by the standard deviation about the mean.  In practice, investors choose a comfortable amount of risk to assume which then translates into an expected return via the efficient frontier.  The efficient frontier is the best one can do in terms of risk-reward efficiency.  It represents the panoramic view of the organizational portfolio depicting the fruition of the highest forecasting and decisionmaking intelligence available.  And while the end result is sufficient reason for conducting the exercise in the first place, the process of going through the analysis is often worthwhile in and of itself.  Chart V6 illustrates second-order risk management by contrasting value creation against the risk associated with value creation—thus ensuring the efficient conversion of uncertain information into value.

#### The Bernoulli Moment Vector (BMV)

The Bernoulli moment vector expands on the notion of the first two statistical moments of the distribution by firstly including M0, which is the value that is exposed to change—and by including the third and fourth moments of the distribution.  M1 is translated into M5 using a utility transformation defined by the Delphi process.  M2-M4 are translated into M6 via the selection of a local confidence level of the portfolio distribution.  The local confidence level is a translation of the global confidence level, which is also defined by the Delphi process.  While M6 represents the lower local confidence bound of the distribution, M7 is the upper local confidence bound.  M8 is the global confidence level of the distribution.  M9 is the fractal coefficient and depicts the coefficient by which risk scales over time.  A value of one-half is the standard Brownian motion coefficient and is equivalent to scaling according to the square root of time.  MM is the correlation coefficient and characterizes the portfolio correlation.  SM represents one iteration of a simulated return.  A description of each element in the BMV follows in the table below.  Below that table is a graphical depiction of the BMV with the BMVdisp switch set to Value, M0 and M2.   Value shows asset values.  M0 shows percentage of exposure.  M2 shows the number of standard deviations.

 MTV MTV is the change in value since the inception of the asset.  The value can be determined externally (ie. the market) or internally via a utility transformation of market value. Return Return is simply the change in value since the inception of an asset.  The value can be determined externally (ie. the market) or internally via a utility transformation of market value. Offset The offset is simply the selected offset for displayed in Charts v2 and v5—and can take on values of None, MTV or Return.  Offset is expressed as either a change in value or a backward rate of return. (display) Expo—M0 Exposure is simply the intuitive concept of initial asset value exposed to change.  For example, five barrels of oil at \$40 per barrel equals \$100 of exposure.  A long position is has positive exposure while a short position is negative. Mu—M1 M1 is the first moment and is the expected or forecasted outcome. SD—M2 M2 or standard deviation or SD is the second moment and is the square root of the mean of the squared deviations of returns from M1 and is equivalent to volatility and the square root variance. Skew—M3 M3 or skewness is the third moment and is calculated as the average of the cubed deviations from the mean, normalized to M2.  M3 is the measure of the asymmetry of a distribution—with a M3 of zero indicating a symmetric distribution.  The normal distribution has a M3 of zero. Kurt—M4 M4 or kurtosis is the fourth moment and is calculated as the average of the fourth-power deviations from the mean, normalized to M2.  M4 is the measure of tail-thickness as well as peakedness of a distribution.  A normal distribution has a M4 of three. VaL—M5 M5 is the translation of M1, possibly using the utility transformation—normalized to M2 VaR—M6 M2-M4 are translated into M6 via the selection of a local confidence level of the portfolio distribution—and is the lower local confidence bound of the distribution. uVaR—M7 M7 is the upper local confidence bound. gVaR—M8 M8 is the global confidence level of the distribution. Frac—M9 M9 is the fractal coefficient and depicts the coefficient by which risk scales over time. Corr—MM MM is the correlation coefficient and characterizes the portfolio correlation—which ranges between minus-one and one.  A value of zero means that the expectation of risk for the portfolio is zero. Sim—SM SM represents one iteration of a simulated return.

##### The Bernoulli Moment Vector (BMV)—Value ##### The Bernoulli Moment Vector (BMV)—Expo—M0 ##### The Bernoulli Moment Vector (BMV)—SD—M2 ## The Bernoulli Micromodels

#### The Market Model

The Bernoulli Market Model (shown above) represents the beachhead for the Bernoulli concept and tracks the analytical process from the inputs of the Delphi, rates and exposure through the technical analysis to the actuarial valuation and decisionmaking.  Advanced forms of the market model include features like intertemporal riskmodeling that involves reproducing data characterized by contemporaneous and intertemporal dependencies—such as energy prices and foreign exchange rates.

#### The Credit Model

The Bernoulli Credit Model starts from a base of the market model with the main point of departure from market risk to credit risk being in the modeling of dynamic exposures.  The general consensus in the literature is that the VaR approach is most appropriate for credit risk management for both pre-settlement and settlement risk.  Pre-settlement risk is a form of credit risk that arises whenever forwards or derivatives are traded.  Settlement risk occurs when there is a non-concurrent exchange of value.  The essence of credit risk involves the modeling of dynamic exposure.

#### The Insurance Model

The Bernoulli Insurance Model uses a garch process to breakdown historical losses into property, business interruption and liability types.  It further breaks down losses into small and large, and then breaks down large losses into frequency and severity.  The model applies Monte Carlo simulation of the frequency and severity distributions against the alternative insurance deductible arrangements and the Delphi results in order to identify the optimal arrangement.

#### The Intertemporal Model

The Bernoulli Intertemporal Model is an approach to modeling data characterized by both intertemporal and contemporaneous dependencies—such as energy prices and foreign exchange rates.  Intertemporal riskmodeling 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 Efficient Frontier Model

The Bernoulli Efficient Frontier Model focuses on optimization—the final of the three basic portfolio algorithms—forecasting, integration and optimization—in constructing an efficient frontier by optimizing for all levels of risk.  The model compares the basic Markowitz Model—method of moments, matrix algebra and linear programming—with the advanced Bernoulli Model—progressive method of moments, Monte Carlo simulation and the Camus distribution, and hill-climbing and genetic algorithms.

#### The Capital Decision Model

The Bernoulli Capital Decision Model delineates the decision of whether Eagle Airlines should buy another airplane—and was inspired by the example found in the book Making Hard Decisions by Clemen.  In addition to tangibly mapping out the capital decisionmaking process, the basic capital decision micromodel also features sensitivity analysis and sets the table for the introduction of decision trees as a fully-integrated component.