Introduction to risk analysis
See also: Risk management introduction, Monte Carlo simulation introduction, ModelRisk functions and windows
Risk analysis is the process of systematically identifying and assessing the potential risks and uncertainties that occur when trying to achieve a certain goal (like reaching a target income or finishing a project), and then finding a feasible strategy for most efficiently controlling those risks.
Why do a risk analysis?
In business and government one faces having to make decisions with uncertain outcome all the time. Understanding the uncertainty can help us make a much better decision.
Imagine that you are a national health care provider considering which of two vaccines to purchase. The two vaccines have the same reported level of efficacy (60%), but further study reveals that there is a difference in confidence attached to these two performance measure: one is twice as uncertain as the other.
All else being equal, the health care provider would purchase the vaccine with the smallest uncertainty about its performance: vaccine A.
Replace vaccine by investment and efficacy by profit and we have a problem in business, for which the answer is the same: pick the investment with the smallest uncertainty, all else being equal (investment A). The principal problem is determining that uncertainty which is the central focus of risk analysis.
We can think of two forms of uncertainty that we have to deal with in risk analysis.
The first is a general sense that the quantity we are trying to estimate has some uncertainty attached to it. This is usually described by a distribution like the one in the figure.
Then we have risk events, which are random events that may or may not occur and for which there is some impact of interest to us.
We can distinguish between two types of events.
A risk is an event that may possibly occur, and if it did occur would have a negative impact on the goals of the organization. Thus a risk is composed of three elements:

The scenario.

Its probability of occurrence.

The size of its impact if it did occur (either a fixed value or a distribution).
An opportunity is an event that may possibly occur, and if it did occur would have a positive impact on the goals of the organization. Thus an opportunity is composed of the same three elements as a risk.
A risk and an opportunity can be considered the opposite sides of the same coin. It is usually easiest to consider a potential event to be a risk if it would have a negative impact and its probability is less than 50%, and if the risk had a probability in excess of 50%, to include it in a base plan and then consider the opportunity of it not occurring.
Moving on from whatif scenarios
Single point or deterministic modelling involves using a single 'best guess' estimate of each variable within a model to determine the model's outcome(s). Sensitivities are then performed on the model to determine how much that outcome might in reality vary from the model outcome. This is achieved by selecting various combinations for each input variable.
These various combinations of possible values around the 'best guess' are commonly known as 'what if' scenarios. The model is often also 'stressed' by putting in values that represent worst case scenarios.
Consider a simple problem that is just the sum of five cost items.
We can use the three points, minimum, best guess and maximum, as values to use in a 'what if' analysis. Since there are five cost items and three values per item, there are 3^{5} = 243 possible 'what if' combinations we could produce! Clearly, this is too large a set of scenarios to have any practical use.
This process suffers from two other important drawbacks:

only three values are being used for each variable, where they could, in fact, take any number of values.

no recognition is being given to the fact that the best guess value is much more likely to occur than the minimum and maximum values.
We can stress the model by adding up the minimum costs to find the best case scenario, and add up the maximum costs to get the worst case scenario, but in doing so the range is usually unrealistically large and offers no real insight. The exception is when the worst case scenario is still acceptable.
Quantitative risk analysis (QRA) using Monte Carlo simulation is similar to 'what if' scenarios in that it generates a number of possible scenarios.
However, it goes one step further by effectively accounting for every possible value that each variable could take and weighting each possible scenario by the probability of its occurrence.
QRA achieves this by modelling each variable within a model by a probability distribution. The structure of a QRA model is usually (there are some important exceptions!) very similar to a deterministic model, with all the multiplications, additions, etc. that link the variables together, except that each variable is represented by a probability distribution function instead of a single value.
The objective of a quantitative risk analysis is to calculate the combined impact of the uncertainty in the model's parameters in order to determine an uncertainty distribution of the possible model outcomes.
Identifying Risks
Risk identification is the first step in a complete risk analysis, given that the objectives of the decision maker have been well defined. There are a number of techniques used to help formalise the identification of risks. This part of a formal risk analysis will often prove to be the most informative and constructive element of the whole process, improving company culture by encouraging greater team effort and reducing blame and should be executed with care.
The organizations participating in a formal risk analysis should take pains to create an open and blameless environment in which expressions of concern and doubt can be openly given.
Brainstorming
Brainstorming is a general technique that can be used for identifying a project's risks, pooling the available information on each risk, and identifying possible risk management options. It involves gathering together a group of project stakeholders under the direction of a neutral and reasonably strongwilled chairperson.
It is prudent to have instructed brainstorm session participants well before the meeting about what one is hoping to achieve, together perhaps with some explanation of the meaning of a 'risk' and an 'opportunity'. They may also have been given prompt lists to think about, or any other means for helping them focus on the task.
Suggesting that one should also consider opportunities adds a certain optimism that balances the rather pessimistic search for risks, although admittedly the ratio to risks to opportunities may well end up in the region of 10:1.
The chairperson's role is to structure the meeting so that all relevant aspects of the project are considered. A prompt list is often useful in this regard. The participants are encouraged to identify risks that they feel could impact on the project. The chairperson tries to ensure that a blameless and honest environment is maintained and that each person is allowed to express his or her opinion regardless of status or personality.
The chairperson may also have to question the group when he or she feels that certain areas are being ignored (this is sometimes not very popular). The group is encouraged to discuss each risk as it is identified and what may be done to reduce its probability and impacts. This aspect of a brainstorming session can be particularly valuable as newly identified risks can often be reduced, eliminated or discounted by agreed actions or extra information supplied from the parties sitting around the table.
Brainstorming sessions are sometimes difficult to organize since they require key (usually very busy) people involved in a project to be in the one place at the same time. They are also expensive in terms of personnel. Minutes from a brainstorming session should be circulated to the participants, so the chairperson needs to keep the group focused. The chairperson (risk analyst) then arranges meetings with each of the individuals to discuss the relevant risks and to collect their estimates of each risk's probability and impacts.
A rather more frank list of risks often appears as a result of these onetoone meetings, especially if as chairperson one collates the identified risks without recording the originator. Eliciting individual estimates of risks is also a very good check on whether there is a consensus.
Prompt Lists
Prompt lists provide a set of categories of risk that are pertinent to the type of project under consideration, or the type of risk being considered by an organization. The lists are used to help people think about and identify risks.
Sometimes different types of lists are used together to further improve the chance of identifying all of the important risks that may occur. For example, in analyzing the risks to some project, one prompt list might look at various aspects of the project (e.g. legal, commercial, technical, etc.) or types of tasks involved in the project (design, construction, testing). A project plan and a work breakdown structure, with all of the major tasks defined, are natural prompt lists. In analyzing the reliability of some manufacturing plant, a list of different types of failure (mechanical, electrical, electronic, human, etc.) or a list of the machines or processes involved could be used.
One could also crosscheck with a plan of the site or a flow diagram of the manufacturing process. Check lists can be used at the same time: these are a series of questions one asks as a result of experience of previous problems or opportune events.
A prompt list will never be exhaustive but acts as a focus of attention in the identification of risks. Whether a risk falls into one category or another is not important, only that the risk is identified. The following list provides an example of a fairly general project prompt list. There will often be a number of subsections for each category:
 Administration
 Project acceptance
 Commercial
 Communication
 Environmental
 Financial
 Knowledge and information
 Legal
 Management
 Partner
 Political
 Quality
 Resources
 Strategic
 Subcontractor
 Technical
The identified risks can then be stored and analyzed in the Pelican risk management software.
10 golden rules
Risk modeling should not be prescriptive, because that inhibits creative thinking, which is essential for solving problems  the fundamental purpose of risk analysis. However, there are a few basic principles that are worth adhering to. Morgan and Henrion (1990) offer excellent "ten golden rules" in relation to quantitative risk and policy analysis. You might want to print these out as a big poster to put on your office's wall.
10 golden rules for risk analysis

The possible responses to risks
The response to correctly identified and evaluated risks generally falls into one of these categories:

Increase! (the project plan may be overly cautious).

Do nothing (because it would cost too much or there is nothing that can be done).

Collect more data (to better understand the risk).

Add a contingency (extra amount to budget, deadline, etc. to allow for possibility of risk).

Reduce (e.g. build in redundancy, take a less risky approach, find ways to reduce the probability or impact).

Share (e.g. with partner or contractor providing they can reasonably handle the impact).

Transfer (e.g. insure, backtoback contract).

Eliminate (e.g. do it another way).

Cancel project.
This list can be helpful in thinking of possible responses to identified risks. It should be borne in mind that these risks responses might in their turn carry secondary risks.
Fallback plans should be developed to deal with risks that are identified and not eliminated. If done well in advance, they can help the organization react efficiently, calmly and in unison in a situation where blame and havoc might normally reign.
ModelRisk
Monte Carlo simulation in Excel. Learn more
Tamara
Adding risk and uncertainty to your project schedule. Learn more
Navigation
 Risk management
 Risk management introduction
 What are risks and opportunities?
 Planning a risk analysis
 Clearly stating risk management questions
 Evaluating risk management options
 Introduction to risk analysis
 The quality of a risk analysis
 Using risk analysis to make better decisions
 Explaining a models assumptions
 Statistical descriptions of model outputs
 Simulation Statistical Results
 Preparing a risk analysis report
 Graphical descriptions of model outputs
 Presenting and using results introduction
 Statistical descriptions of model results
 Mean deviation (MD)
 Range
 Semivariance and semistandard deviation
 Kurtosis (K)
 Mean
 Skewness (S)
 Conditional mean
 Custom simulation statistics table
 Mode
 Cumulative percentiles
 Median
 Relative positioning of mode median and mean
 Variance
 Standard deviation
 Interpercentile range
 Normalized measures of spread  the CofV
 Graphical descriptionss of model results
 Showing probability ranges
 Overlaying histogram plots
 Scatter plots
 Effect of varying number of bars
 Sturges rule
 Relationship between cdf and density (histogram) plots
 Difficulty of interpreting the vertical scale
 Stochastic dominance tests
 Riskreturn plots
 Second order cumulative probability plot
 Ascending and descending cumulative plots
 Tornado plot
 Box Plot
 Cumulative distribution function (cdf)
 Probability density function (pdf)
 Crude sensitivity analysis for identifying important input distributions
 Pareto Plot
 Trend plot
 Probability mass function (pmf)
 Overlaying cdf plots
 Cumulative Plot
 Simulation data table
 Statistics table
 Histogram Plot
 Spider plot
 Determining the width of histogram bars
 Plotting a variable with discrete and continuous elements
 Smoothing a histogram plot
 Risk analysis modeling techniques
 Monte Carlo simulation
 Monte Carlo simulation introduction
 Monte Carlo simulation in ModelRisk
 Filtering simulation results
 Output/Input Window
 Simulation Progress control
 Running multiple simulations
 Random number generation in ModelRisk
 Random sampling from input distributions
 How many Monte Carlo samples are enough?
 Probability distributions
 Distributions introduction
 Probability calculations in ModelRisk
 Selecting the appropriate distributions for your model
 List of distributions by category
 Distribution functions and the U parameter
 Univariate continuous distributions
 Beta distribution
 Beta Subjective distribution
 Fourparameter Beta distribution
 Bradford distribution
 Burr distribution
 Cauchy distribution
 Chi distribution
 Chi Squared distribution
 Continuous distributions introduction
 Continuous fitted distribution
 Cumulative ascending distribution
 Cumulative descending distribution
 Dagum distribution
 Erlang distribution
 Error distribution
 Error function distribution
 Exponential distribution
 Exponential family of distributions
 Extreme Value Minimum distribution
 Extreme Value Maximum distribution
 F distribution
 Fatigue Life distribution
 Gamma distribution
 Generalized Extreme Value distribution
 Generalized Logistic distribution
 Generalized Trapezoid Uniform (GTU) distribution
 Histogram distribution
 HyperbolicSecant distribution
 Inverse Gaussian distribution
 Johnson Bounded distribution
 Johnson Unbounded distribution
 Kernel Continuous Unbounded distribution
 Kumaraswamy distribution
 Kumaraswamy Fourparameter distribution
 Laplace distribution
 Levy distribution
 Lifetime TwoParameter distribution
 Lifetime ThreeParameter distribution
 Lifetime Exponential distribution
 LogGamma distribution
 Logistic distribution
 LogLaplace distribution
 LogLogistic distribution
 LogLogistic Alternative parameter distribution
 LogNormal distribution
 LogNormal Alternativeparameter distribution
 LogNormal base B distribution
 LogNormal base E distribution
 LogTriangle distribution
 LogUniform distribution
 Noncentral Chi squared distribution
 Noncentral F distribution
 Normal distribution
 Normal distribution with alternative parameters
 Maxwell distribution
 Normal Mix distribution
 Relative distribution
 Ogive distribution
 Pareto (first kind) distribution
 Pareto (second kind) distribution
 Pearson Type 5 distribution
 Pearson Type 6 distribution
 Modified PERT distribution
 PERT distribution
 PERT Alternativeparameter distribution
 Reciprocal distribution
 Rayleigh distribution
 Skew Normal distribution
 Slash distribution
 SplitTriangle distribution
 Studentt distribution
 Threeparameter Student distribution
 Triangle distribution
 Triangle Alternativeparameter distribution
 Uniform distribution
 Weibull distribution
 Weibull Alternativeparameter distribution
 ThreeParameter Weibull distribution
 Univariate discrete distributions
 Discrete distributions introduction
 Bernoulli distribution
 BetaBinomial distribution
 BetaGeometric distribution
 BetaNegative Binomial distribution
 Binomial distribution
 Burnt Finger Poisson distribution
 Delaporte distribution
 Discrete distribution
 Discrete Fitted distribution
 Discrete Uniform distribution
 Geometric distribution
 HypergeoM distribution
 Hypergeometric distribution
 HypergeoD distribution
 Inverse Hypergeometric distribution
 Logarithmic distribution
 Negative Binomial distribution
 Poisson distribution
 Poisson Uniform distribution
 Polya distribution
 Skellam distribution
 Step Uniform distribution
 Zeromodified counting distributions
 More on probability distributions
 Multivariate distributions
 Multivariate distributions introduction
 Dirichlet distribution
 Multinomial distribution
 Multivariate Hypergeometric distribution
 Multivariate Inverse Hypergeometric distribution type2
 Negative Multinomial distribution type 1
 Negative Multinomial distribution type 2
 Multivariate Inverse Hypergeometric distribution type1
 Multivariate Normal distribution
 More on probability distributions
 Approximating one distribution with another
 Approximations to the Inverse Hypergeometric Distribution
 Normal approximation to the Gamma Distribution
 Normal approximation to the Poisson Distribution
 Approximations to the Hypergeometric Distribution
 Stirlings formula for factorials
 Normal approximation to the Beta Distribution
 Approximation of one distribution with another
 Approximations to the Negative Binomial Distribution
 Normal approximation to the Studentt Distribution
 Approximations to the Binomial Distribution
 Normal_approximation_to_the_Binomial_distribution
 Poisson_approximation_to_the_Binomial_distribution
 Normal approximation to the Chi Squared Distribution
 Recursive formulas for discrete distributions
 Normal approximation to the Lognormal Distribution
 Normal approximations to other distributions
 Approximating one distribution with another
 Correlation modeling in risk analysis
 Common mistakes when adapting spreadsheet models for risk analysis
 More advanced risk analysis methods
 SIDs
 Modeling with objects
 ModelRisk database connectivity functions
 PK/PD modeling
 Value of information techniques
 Simulating with ordinary differential equations (ODEs)
 Optimization of stochastic models
 ModelRisk optimization extension introduction
 Optimization Settings
 Defining Simulation Requirements in an Optimization Model
 Defining Decision Constraints in an Optimization Model
 Optimization Progress control
 Defining Targets in an Optimization Model
 Defining Decision Variables in an Optimization Model
 Optimization Results
 Summing random variables
 Aggregate distributions introduction
 Aggregate modeling  Panjer's recursive method
 Adding correlation in aggregate calculations
 Sum of a random number of random variables
 Moments of an aggregate distribution
 Aggregate modeling in ModelRisk
 Aggregate modeling  Fast Fourier Transform (FFT) method
 How many random variables add up to a fixed total
 Aggregate modeling  compound Poisson approximation
 Aggregate modeling  De Pril's recursive method
 Testing and modeling causal relationships
 Stochastic time series
 Time series introduction
 Time series in ModelRisk
 Autoregressive models
 Thiel inequality coefficient
 Effect of an intervention at some uncertain point in time
 Log return of a Time Series
 Markov Chain models
 Seasonal time series
 Bounded random walk
 Time series modeling in finance
 Birth and death models
 Time series models with leading indicators
 Geometric Brownian Motion models
 Time series projection of events occurring randomly in time
 Simulation for six sigma
 ModelRisk's Six Sigma functions
 VoseSixSigmaCp
 VoseSixSigmaCpkLower
 VoseSixSigmaProbDefectShift
 VoseSixSigmaLowerBound
 VoseSixSigmaK
 VoseSixSigmaDefectShiftPPMUpper
 VoseSixSigmaDefectShiftPPMLower
 VoseSixSigmaDefectShiftPPM
 VoseSixSigmaCpm
 VoseSixSigmaSigmaLevel
 VoseSixSigmaCpkUpper
 VoseSixSigmaCpk
 VoseSixSigmaDefectPPM
 VoseSixSigmaProbDefectShiftLower
 VoseSixSigmaProbDefectShiftUpper
 VoseSixSigmaYield
 VoseSixSigmaUpperBound
 VoseSixSigmaZupper
 VoseSixSigmaZmin
 VoseSixSigmaZlower
 Modeling expert opinion
 Modeling expert opinion introduction
 Sources of error in subjective estimation
 Disaggregation
 Distributions used in modeling expert opinion
 A subjective estimate of a discrete quantity
 Incorporating differences in expert opinions
 Modeling opinion of a variable that covers several orders of magnitude
 Maximum entropy
 Probability theory and statistics
 Probability theory and statistics introduction
 Stochastic processes
 Stochastic processes introduction
 Poisson process
 Hypergeometric process
 The hypergeometric process
 Number in a sample with a particular characteristic in a hypergeometric process
 Number of hypergeometric samples to get a specific number of successes
 Number of samples taken to have an observed s in a hypergeometric process
 Estimate of population and subpopulation sizes in a hypergeometric process
 The binomial process
 Renewal processes
 Mixture processes
 Martingales
 Estimating model parameters from data
 The basics
 Probability equations
 Probability theorems and useful concepts
 Probability parameters
 Probability rules and diagrams
 The definition of probability
 The basics of probability theory introduction
 Fitting probability models to data
 Fitting time series models to data
 Fitting correlation structures to data
 Fitting in ModelRisk
 Fitting probability distributions to data
 Fitting distributions to data
 Method of Moments (MoM)
 Check the quality of your data
 KolmogorovSmirnoff (KS) Statistic
 AndersonDarling (AD) Statistic
 Goodness of fit statistics
 The ChiSquared GoodnessofFit Statistic
 Determining the joint uncertainty distribution for parameters of a distribution
 Using Method of Moments with the Bootstrap
 Maximum Likelihood Estimates (MLEs)
 Fitting a distribution to truncated censored or binned data
 Critical Values and Confidence Intervals for GoodnessofFit Statistics
 Matching the properties of the variable and distribution
 Transforming discrete data before performing a parametric distribution fit
 Does a parametric distribution exist that is well known to fit this type of variable?
 Censored data
 Fitting a continuous nonparametric secondorder distribution to data
 Goodness of Fit Plots
 Fitting a second order Normal distribution to data
 Using Goodnessof Fit Statistics to optimize Distribution Fitting
 Information criteria  SIC HQIC and AIC
 Fitting a second order parametric distribution to observed data
 Fitting a distribution for a continuous variable
 Does the random variable follow a stochastic process with a wellknown model?
 Fitting a distribution for a discrete variable
 Fitting a discrete nonparametric secondorder distribution to data
 Fitting a continuous nonparametric firstorder distribution to data
 Fitting a first order parametric distribution to observed data
 Fitting a discrete nonparametric firstorder distribution to data
 Fitting distributions to data
 Technical subjects
 Comparison of Classical and Bayesian methods
 Comparison of classic and Bayesian estimate of Normal distribution parameters
 Comparison of classic and Bayesian estimate of intensity lambda in a Poisson process
 Comparison of classic and Bayesian estimate of probability p in a binomial process
 Which technique should you use?
 Comparison of classic and Bayesian estimate of mean "time" beta in a Poisson process
 Classical statistics
 Bayesian
 Bootstrap
 The Bootstrap
 Linear regression parametric Bootstrap
 The Jackknife
 Multiple variables Bootstrap Example 2: Difference between two population means
 Linear regression nonparametric Bootstrap
 The parametric Bootstrap
 Bootstrap estimate of prevalence
 Estimating parameters for multiple variables
 Example: Parametric Bootstrap estimate of the mean of a Normal distribution with known standard deviation
 The nonparametric Bootstrap
 Example: Parametric Bootstrap estimate of mean number of calls per hour at a telephone exchange
 The Bootstrap likelihood function for Bayesian inference
 Multiple variables Bootstrap Example 1: Estimate of regression parameters
 Bayesian inference
 Uninformed priors
 Conjugate priors
 Prior distributions
 Bayesian analysis with threshold data
 Bayesian analysis example: gender of a random sample of people
 Informed prior
 Simulating a Bayesian inference calculation
 Hyperparameters
 Hyperparameter example: Microfractures on turbine blades
 Constructing a Bayesian inference posterior distribution in Excel
 Bayesian analysis example: Tigers in the jungle
 Markov chain Monte Carlo (MCMC) simulation
 Introduction to Bayesian inference concepts
 Bayesian estimate of the mean of a Normal distribution with known standard deviation
 Bayesian estimate of the mean of a Normal distribution with unknown standard deviation
 Determining prior distributions for correlated parameters
 Improper priors
 The Jacobian transformation
 Subjective prior based on data
 Taylor series approximation to a Bayesian posterior distribution
 Bayesian analysis example: The Monty Hall problem
 Determining prior distributions for uncorrelated parameters
 Subjective priors
 Normal approximation to the Beta posterior distribution
 Bayesian analysis example: identifying a weighted coin
 Bayesian estimate of the standard deviation of a Normal distribution with known mean
 Likelihood functions
 Bayesian estimate of the standard deviation of a Normal distribution with unknown mean
 Determining a prior distribution for a single parameter estimate
 Simulating from a constructed posterior distribution
 Bootstrap
 Comparison of Classical and Bayesian methods
 Analyzing and using data introduction
 Data Object
 Vose probability calculation
 Bayesian model averaging
 Miscellaneous
 Excel and ModelRisk model design and validation techniques
 Using range names for model clarity
 Color coding models for clarity
 Compare with known answers
 Checking units propagate correctly
 Stressing parameter values
 Model Validation and behavior introduction
 Informal auditing
 Analyzing outputs
 View random scenarios on screen and check for credibility
 Split up complex formulas (megaformulas)
 Building models that are efficient
 Comparing predictions against reality
 Numerical integration
 Comparing results of alternative models
 Building models that are easy to check and modify
 Model errors
 Model design introduction
 About array functions in Excel
 Excel and ModelRisk model design and validation techniques
 Monte Carlo simulation
 RISK ANALYSIS SOFTWARE
 Risk analysis software from Vose Software
 ModelRisk  risk modeling in Excel
 ModelRisk functions explained
 VoseCopulaOptimalFit and related functions
 VoseTimeOptimalFit and related functions
 VoseOptimalFit and related functions
 VoseXBounds
 VoseCLTSum
 VoseAggregateMoments
 VoseRawMoments
 VoseSkewness
 VoseMoments
 VoseKurtosis
 VoseAggregatePanjer
 VoseAggregateFFT
 VoseCombined
 VoseCopulaBiGumbel
 VoseCopulaBiClayton
 VoseCopulaBiNormal
 VoseCopulaBiT
 VoseKendallsTau
 VoseRiskEvent
 VoseCopulaBiFrank
 VoseCorrMatrix
 VoseRank
 VoseValidCorrmat
 VoseSpearman
 VoseCopulaData
 VoseCorrMatrixU
 VoseTimeSeasonalGBM
 VoseMarkovSample
 VoseMarkovMatrix
 VoseThielU
 VoseTimeEGARCH
 VoseTimeAPARCH
 VoseTimeARMA
 VoseTimeDeath
 VoseTimeAR1
 VoseTimeAR2
 VoseTimeARCH
 VoseTimeMA2
 VoseTimeGARCH
 VoseTimeGBMJDMR
 VoseTimePriceInflation
 VoseTimeGBMMR
 VoseTimeWageInflation
 VoseTimeLongTermInterestRate
 VoseTimeMA1
 VoseTimeGBM
 VoseTimeGBMJD
 VoseTimeShareYields
 VoseTimeYule
 VoseTimeShortTermInterestRate
 VoseDominance
 VoseLargest
 VoseSmallest
 VoseShift
 VoseStopSum
 VoseEigenValues
 VosePrincipleEsscher
 VoseAggregateMultiFFT
 VosePrincipleEV
 VoseCopulaMultiNormal
 VoseRunoff
 VosePrincipleRA
 VoseSumProduct
 VosePrincipleStdev
 VosePoissonLambda
 VoseBinomialP
 VosePBounds
 VoseAIC
 VoseHQIC
 VoseSIC
 VoseOgive1
 VoseFrequency
 VoseOgive2
 VoseNBootStdev
 VoseNBoot
 VoseSimulate
 VoseNBootPaired
 VoseAggregateMC
 VoseMean
 VoseStDev
 VoseAggregateMultiMoments
 VoseDeduct
 VoseExpression
 VoseLargestSet
 VoseKthSmallest
 VoseSmallestSet
 VoseKthLargest
 VoseNBootCofV
 VoseNBootPercentile
 VoseExtremeRange
 VoseNBootKurt
 VoseCopulaMultiClayton
 VoseNBootMean
 VoseTangentPortfolio
 VoseNBootVariance
 VoseNBootSkewness
 VoseIntegrate
 VoseInterpolate
 VoseCopulaMultiGumbel
 VoseCopulaMultiT
 VoseAggregateMultiMC
 VoseCopulaMultiFrank
 VoseTimeMultiMA1
 VoseTimeMultiMA2
 VoseTimeMultiGBM
 VoseTimeMultBEKK
 VoseAggregateDePril
 VoseTimeMultiAR1
 VoseTimeWilkie
 VoseTimeDividends
 VoseTimeMultiAR2
 VoseRuinFlag
 VoseRuinTime
 VoseDepletionShortfall
 VoseDepletion
 VoseDepletionFlag
 VoseDepletionTime
 VosejProduct
 VoseCholesky
 VoseTimeSimulate
 VoseNBootSeries
 VosejkProduct
 VoseRuinSeverity
 VoseRuin
 VosejkSum
 VoseTimeDividendsA
 VoseRuinNPV
 VoseTruncData
 VoseSample
 VoseIdentity
 VoseCopulaSimulate
 VoseSortA
 VoseFrequencyCumulA
 VoseAggregateDeduct
 VoseMeanExcessP
 VoseProb10
 VoseSpearmanU
 VoseSortD
 VoseFrequencyCumulD
 VoseRuinMaxSeverity
 VoseMeanExcessX
 VoseRawMoment3
 VosejSum
 VoseRawMoment4
 VoseNBootMoments
 VoseVariance
 VoseTimeShortTermInterestRateA
 VoseTimeLongTermInterestRateA
 VoseProb
 VoseDescription
 VoseCofV
 VoseAggregateProduct
 VoseEigenVectors
 VoseTimeWageInflationA
 VoseRawMoment1
 VosejSumInf
 VoseRawMoment2
 VoseShuffle
 VoseRollingStats
 VoseSplice
 VoseTSEmpiricalFit
 VoseTimeShareYieldsA
 VoseParameters
 VoseAggregateTranche
 VoseCovToCorr
 VoseCorrToCov
 VoseLLH
 VoseTimeSMEThreePoint
 VoseDataObject
 VoseCopulaDataSeries
 VoseDataRow
 VoseDataMin
 VoseDataMax
 VoseTimeSME2Perc
 VoseTimeSMEUniform
 VoseTimeSMESaturation
 VoseOutput
 VoseInput
 VoseTimeSMEPoisson
 VoseTimeBMAObject
 VoseBMAObject
 VoseBMAProb10
 VoseBMAProb
 VoseCopulaBMA
 VoseCopulaBMAObject
 VoseTimeEmpiricalFit
 VoseTimeBMA
 VoseBMA
 VoseSimKurtosis
 VoseOptConstraintMin
 VoseSimProbability
 VoseCurrentSample
 VoseCurrentSim
 VoseLibAssumption
 VoseLibReference
 VoseSimMoments
 VoseOptConstraintMax
 VoseSimMean
 VoseOptDecisionContinuous
 VoseOptRequirementEquals
 VoseOptRequirementMax
 VoseOptRequirementMin
 VoseOptTargetMinimize
 VoseOptConstraintEquals
 VoseSimVariance
 VoseSimSkewness
 VoseSimTable
 VoseSimCofV
 VoseSimPercentile
 VoseSimStDev
 VoseOptTargetValue
 VoseOptTargetMaximize
 VoseOptDecisionDiscrete
 VoseSimMSE
 VoseMin
 VoseMin
 VoseOptDecisionList
 VoseOptDecisionBoolean
 VoseOptRequirementBetween
 VoseOptConstraintBetween
 VoseSimMax
 VoseSimSemiVariance
 VoseSimSemiStdev
 VoseSimMeanDeviation
 VoseSimMin
 VoseSimCVARp
 VoseSimCVARx
 VoseSimCorrelation
 VoseSimCorrelationMatrix
 VoseOptConstraintString
 VoseOptCVARx
 VoseOptCVARp
 VoseOptPercentile
 VoseSimValue
 VoseSimStop
 Precision Control Functions
 VoseAggregateDiscrete
 VoseTimeMultiGARCH
 VoseTimeGBMVR
 VoseTimeGBMAJ
 VoseTimeGBMAJVR
 VoseSID
 Generalized Pareto Distribution (GPD)
 Generalized Pareto Distribution (GPD) Equations
 ThreePoint Estimate Distribution
 ThreePoint Estimate Distribution Equations
 VoseCalibrate
 ModelRisk interfaces
 Integrate
 Data Viewer
 Stochastic Dominance
 Library
 Correlation Matrix
 Portfolio Optimization Model
 Common elements of ModelRisk interfaces
 Risk Event
 Extreme Values
 Select Distribution
 Combined Distribution
 Aggregate Panjer
 Interpolate
 View Function
 Find Function
 Deduct
 Ogive
 AtRISK model converter
 Aggregate Multi FFT
 Stop Sum
 Crystal Ball model converter
 Aggregate Monte Carlo
 Splicing Distributions
 Subject Matter Expert (SME) Time Series Forecasts
 Aggregate Multivariate Monte Carlo
 Ordinary Differential Equation tool
 Aggregate FFT
 More on Conversion
 Multivariate Copula
 Bivariate Copula
 Univariate Time Series
 Modeling expert opinion in ModelRisk
 Multivariate Time Series
 Sum Product
 Aggregate DePril
 Aggregate Discrete
 Expert
 ModelRisk introduction
 Building and running a simple example model
 Distributions in ModelRisk
 List of all ModelRisk functions
 Custom applications and macros
 ModelRisk functions explained
 Tamara  project risk analysis
 Introduction to Tamara project risk analysis software
 Launching Tamara
 Importing a schedule
 Assigning uncertainty to the amount of work in the project
 Assigning uncertainty to productivity levels in the project
 Adding risk events to the project schedule
 Adding cost uncertainty to the project schedule
 Saving the Tamara model
 Running a Monte Carlo simulation in Tamara
 Reviewing the simulation results in Tamara
 Using Tamara results for cost and financial risk analysis
 Creating, updating and distributing a Tamara report
 Tips for creating a schedule model suitable for Monte Carlo simulation
 Random number generator and sampling algorithms used in Tamara
 Probability distributions used in Tamara
 Correlation with project schedule risk analysis
 Pelican  enterprise risk management
 ModelRisk Cloud system
 ModelRisk Cloud introduction
 Getting your software ready
 Starting ModelRisk Cloud
 Uploading a risk analysis model
 Creating a new scenario for the risk analysis model
 Running a Monte Carlo simulation of the model
 Uploading a SID (Simulation Imported Data file)
 Building a risk analysis model that uses SIDs
 Viewing the Monte Carlo results from a simulation run
 Administrator's use of ModelRisk Cloud
 Preparing a risk analysis model for upload to ModelRisk Cloud
 ModelRisk Result Viewer