Discounted cash flow modeling
See also: Insurance and finance risk analysis modeling introduction, Time series modeling in finance, Aggregate distributions introduction, Modeling with objects
First have a look at the following diagram:
A typical discounted cashflow model for a potential investment makes forecasts of costs and revenues over the life of the project and discounts those revenues back to a present value. Most analysts start with a 'base case' model and add uncertainty to the important elements of the model. Happily, the mathematics involved in adding risk to these types of models is quite simple. In this topic we assume that you can build a base case cashflow model that will look something like the example model shown below. We will then focus on the input modelling elements and some financial outputs.
Example model NPV of a capital investment (tab Static model)  a typical, if somewhat reduced, cash flow model.
There are a number of relevant topics that are covered in other parts of this help file. Please make sure you have a good understanding of these first:
In capital investment models we rely a great deal on expert judgement to estimate variables like costs, time to market, sales volumes, discount levels
We don't usually have a great deal of historic data to work with in capital investment projects because the investment is new. I have worked with a very successful retail company that investigates levels of pedestrian traffic at different locations in a town where it is considering locating a new outlet. It has excellent regional data on how that traffic converts to till receipts. That is quite typical of the type of data one might have for a cashflow analysis and we will go through such a model later in this topic. Hydrocarbon and mineral exploration will generally have improving levels of data about the reserves, but have specialised methods for statistically analyzing (e.g. Krieging) their data, so I won't consider them further here.
Simple forms of correlation modelling  recognizing that two or more variables are likely to be linked in some way  are very important in cashflow models.
The time series topics deal with many different technical time series models. GBM, seasonal and autoregressive models are useful for modelling inflation, exchange and interest rates over time in a cashflow model. Lead indicators can help predict market size a short time into the future. In this topic consider variables like demand for products and sales volumes which are generally built on a more intuitive basis.
Common errors in risk modeling
Risk analysis cashflow models are not generally that technically complicated, but our reviews show that the types of errors described under Common errors in risk modeling appear very frequently, so we encourage you to read that section carefully. The rest of this topic offers some ideas on model building that are very applicable to cashflow models.
Useful time series models of sales and market size
Effect of an intervention at some uncertain point in time
Time series variables are often affected by single identifiable 'shocks', like elections, changes to a law, introduction of a competitor, start or finish of a war, a scandal, etc. The modelling of the occurrence of a shock and its effects may need to take into account several elements:

When the shock may occur (this could be random);

Whether this changes the probability or impact of other possible shocks;

The effect of the shock: magnitude and duration
Consider the following problem: People are purchasing your product at a current rate of 88/month, and the rate appears to be increasing by 1.3 sales/month with each month. However, we are 80% sure that a competitor is going to enter the market and will do so between 20 and 50 months from now. If the competitor enters the market they will take about 30% of your sales. Forecast the number of sales there will be for the next 100 months.
Possible pathways generated by the model depending on whether the competitor enters the market
The model for this problem is shown in the figure above . The Bernoulli variable returns a 1 with 80% probability, otherwise a 0. It is used as a 'flag', the 1 representing a competitor entry, the 0 representing no competitor. Other cells use conditional logic to adapt to the scenario.
The StepUniform generates integer values between 20 and 50, and cell E4 returns the month 1000 if the competitor does not enter the market i.e. a time beyond the modelled period. It is a good idea if you use this type of technique to make such a number very far from the range of the modelled period in case someone decided to extend the period analyzed. A Poisson distribution is used to model the number of sales reflecting that the sales are independent of each other and randomly distributed in time. The nice thing about a Poisson distribution is that it takes just one parameter  its mean, so you don't have to think about variation about that mean separately (e.g. determine a standard deviation).
Example model Poisson_sales_with_competitor  Poisson sales affected by the possible entry of a competitor
Distributing market share
When competitors enter an established market they have to establish the reputation of their product and fight for market share with others that are already established. This takes time, so it is more realistic to model a gradual loss of market share to competitors.
Consider the following problem: Market volume for your product is expected to grow each year by (10%, 20%, 40%) beginning next year at (2500, 3000, 5000) up to a maximum of 20,000 units. You expect one competitor to emerge as soon as the market volume reaches 3,500 units in the previous year. A second would appear at 8,500 units. Your competitors' shares of the market would grow linearly until you all have equal market share after three years. Model the sales you will make.
Example model New_competitors  sales where the total market is divided with new entry competitors
The figure above shows the model. It is mostly selfexplanatory. The interesting component lies in Cells F10:L10 which divides the forecast market for your product among the average of the number of competitors over the last three years and yourself (the '1' in the equation). Averaging over three years is a neat way of allocating an emerging competitor 1/3 of your market strength in the first year, 2/3 in the second and equal strength from the third year on  meaning that they will then sell as many units as you. What is so helpful about this little trick is that it automatically takes into account each new competitor and when they entered the market, which is rather difficult to do otherwise. Note that we need three zeros in cells C8:E8 to initialise the model.
Reduced sales over time to a finite market
Some products are essentially a once in a lifetime purchase, e.g. a life insurance, big flat screen TV, a new guttering system and a pet identification chip. If we are initially quite successful in selling the product into the potential market, the remaining market size decreases although this can be compensated to some degree by new potential consumers. Consider the following problem: There are currently PERT(50000,55000,60000) possible purchasers of your product. Each year there will be about a 10% turnover (meaning 10% more possible purchasers will appear). The probability that you will sell to any particular purchaser in a year is PERT(10%,20%,35%). Forecast sales for the next 10 years.
Example model Finite_market  forecasting sales over time to a finite market
The figure above shows the model for this problem. Note that C8:C16 is subtracting sales already made from the previous year's market size but also adding in a regenerated market element. The Binomial distribution then converts the current market size to sales. In the particular scenario shown in the figure the probability of selling is high (26%) so sales start off high and drop off quickly since the regeneration rate is so much lower (10%). Note that some Monte Carlo software cannot handle large numbers of trials in their Binomial distribution in which case you will need to use a Poisson or Normal approximation.
Growth of sales over time up to a maximum as a function of marketing effort
Sometimes we might find it easier to estimate what our annual sales will be when stabilized, but be unsure of how quickly we will be able to achieve that stability. In this sort of situation it can be easier to model a theoretical maximum sales and match it to some ramping function. A typical form of such a ramping function r(t) is:
which will produce a curve that starts at 0 for t = 0 and asymptotically reaches 1 at an infinite value of t, but reaches 0.5 at . Consider the following problem: you expect a final sales rate of PERT(1800,2300,3600) and expect to achieve half that in the next PERT(3.5,4,5) years. Produce a sales forecast for the next 10 years.
The example model below provides a solution:
Example model Sales_with_maximum  forecasting ramping sales to an uncertain theoretical maximum
Summing random variables
Perhaps the most common errors in cashflow modelling occur when one wishes to sum a number of random costs, sales or revenues. For example, imagine that you expect to have Lognormal(100 000, 25 000) customers enter your store per year and they will spend $Lognormal(55, 12) each  how would you estimate the total revenue? People generally write something like:
= ROUND(Lognormal(100 000, 25 000),0) * $Lognormal(55, 12) (equation 1)
using the ROUND function in Excel to recognise that the number of people must be discrete. But let's think what happens when the software starts simulating. It will pick a random value from each distribution and multiply them together. Picking a reasonably high till receipt, the probability that a random customer will spend more than $70, for example, is:
=1VoseLognormalProb(70,55,12,1) = 0.11%
The probability that two people will do the same is 11% * 11% = 1.2%, and the probability that thousands of people will spend that much is infinitesimally small. However, Equation 1 will assign a 11% probability that all customers will spend over $70 no matter how many there are. The equation is wrong because it should have summed ROUND(Lognormal(100 000, 25 000),0) separate Lognormal(55, 12) distributions. That's a big, slow model so we use a variety of techniques to shortcut to the answer, which is the topic of aggregate modeling.
Summing variable margins on variable revenues
A common situation is that we have a large random number of revenue items that follow the same probability distribution but which are independent of each other, and we have independent profit margins that follow another distribution that must be applied to each revenue item. This type of model quickly becomes extremely cumbersome to model because for each revenue item we need two distributions: one for revenue and another for the profit margin and we may have many types of large numbers of revenue items. It is such a common problem that we designed a function in ModelRisk to handle this, allowing you to keep the model to a manageable size, speeding up simulation time, and making the model far simpler to review. Perhaps most importantly, it allows you to avoid a lot of conditional logic that is easy to get wrong. Consider the following problem: a capital venture company is considering investing in a company that makes TV shows. They expect to make PERT(28,32,39) pilots next year which will generate revenues of $PERT(120,150,250)k each independently and from which the profit margin is PERT(1%,5%,12%). There is a 30% chance that each pilot is made into a TV series run in that country running for Discrete({1,2,3,4,5},{0.4,0.25,0.2,0.1,0.05}) series, where each season of each series generates $PERT(120,150,250)k with margins of PERT(15%,25%,45%). There is a 20% chance that these local series would be sold to the US generating $PERT(240,255,1350) per season sold of which the profit margin is PERT(65%,70%,85%). What is the total profit generated from next year's pilots?
Example model TV_series_profits  forecasting profits from a TV series
The problem is not technically difficult but the scale of the modelling explodes very quickly. We worked on the model for a real investment of this type and it had many more layers: pilots in several countries, merchandising of various types, repeats, etc. and it took a lot of effort to manage. The figure shows how using objects for modeling allows us a surprisingly succinct model: rows 2 to 12 are the input data, rows 14 to 16 are the actual calculations.
There are a few things to point out. In Cell F2, ½ is subtracted and added to the minimum and maximum estimates respectively of the number of pilots to give a more realistic chance of their occurrence after rounding. Distributions are input as ModelRisk objects in cells F3, F4, F6, F7, F8, F10 and F11 because we want to use these distributions many times. Cell C16 and elsewhere uses the VoseSumProduct function to add together revenue * margin for each pilot where the revenue and margin distributions are defined by the distribution Objects in Cells F3 and F4 respectively. Cell F14 simulates the number of pilots that made it to become series, from which the model determines how many of those become series also sold into the US in Cell E14, the difference being the number of pilots that only became local series in Cell D14. Setting up the logic this ways ensures that we have a consistent model: the local only and US & local series always add up to the total series produced. Cells D15 and E15 use the VoseAggregateMC(x, y) function to simulate the sum of x random variables all taking the same distribution y defined as an Object.
Financial measures in risk analysis
The two main measures of profitability in DCF models are net present value (NPV) and internal rate of return (IRR). The two main measures of financial exposure are VAR and expected shortfall. Their pros and cons are discussed below.
Net Present Value
Net Present Value (NPV) attempts to determine the present value of a series of cashflows from a project that stretches out into the future. This present value is a measure of how much the company is gaining at today's money by undertaking the project: in other words, how much more the company itself will be worth by accepting the project.
An NPV calculation discounts future cashflows at a specified discount rate r that takes account of:

the time value of money (e.g. if inflation is running at 4%, £1.04 in a year's time is only worth £1.00 today);

the interest that could have been earned over inflation by investing instead in a guaranteed investment;

the extra return that is required over (1) and (2) to compensate for the degree of risk that is being accepted in this project;
Parts (1) and (2) are combined to produce the risk free interest rate, r_{f}. This is typically determined as the interest paid by guaranteed fixed payment investments like government bonds with a term roughly equivalent to the duration of the project.
The extra interest r^{*} over r_{f} needed for (3) is determined by looking at the uncertainty of the project. In risk analysis models, this uncertainty is represented by the spread of the distributions of cashflow for each period. The sum of r^{*} and r_{f} is called the riskadjusted discount rate r.
The most commonly used calculation for the NPV of a cashflow series over n periods is as follows:
where are the expected (i.e. average) values of the cashflows in each period and r is the riskadjusted discount rate.
NPV calculations performed in a risk analysis spreadsheet model are usually presented as a distribution of NPVs because the cashflow values selected in the NPV calculations are their distributions rather than their expected values. Theoretically, this is incorrect. Since an NPV is the net present value, it can have no uncertainty. It is the amount of money that the company values the project at today. The problem is that we have double counted our risk by first discounting at the riskadjusted discounted rate r and then showing the NPV as a distribution (i.e. it is uncertain).
Two theoretically correct methods for calculating an NPV in risk analysis are discussed below, along with a more practical, but strictly speaking incorrect, alternative:
Theoretical approach 1: Discount the cashflow distributions at the risk free rate.
This produces a distribution of NPVs at r_{f} and ensures that the risk is not doublecounted. However, such a distribution is not at all easy to interpret since decision makers will almost certainly never have dealt with risk free rate NPVs and therefore have nothing to compare the model output against.
Theoretical approach 2: Discount the expected value of each cashflow at the riskadjusted discount rate.
This is the application of the above formula. It results in a single figure for the NPV of the project. A risk analysis is run to determine the expected value and spread of the cashflows in each period. The discount rate is usually determined by comparing the riskiness associated with the project's cashflows against the riskiness of other projects in the company's portfolio. The company can then assign a discount rate above or below its usual discount rate depending on whether the project being analyzedexhibits more or less risk than the average. Some companies determine a range of discount rates (three or so) to be used against projects of different riskiness.
The major problems of this method are that it assumes the cashflow distributions are symmetric and that no correlation exists between cashflows. Distributions of costs and returns almost always exhibit some form of asymmetry and in a typical investment project there is also always some form of correlation between cashflow periods: for example, sales in one period will be affected by previous sales, a capital injection in one period often means that it doesn't occur in the next one (e.g. expansion of a factory) or the model may include a time series forecast of prices, production rates or sales volume that are autocorrelated. If there is a strong positive correlation between cashflows, this method will overestimate the NPV. Conversely, a strong negative correlation between cashflows will result in the NPV being underestimated. The correlation between cashflows may take any number of, often complex, forms. Financial theory does not offer a practical method for adjusting the NPV to take account of these correlations.
In practice, it is easier to apply the riskadjusted discount rate r to the cashflow distributions to produce a distribution of NPVs. This method incorporates correlation between distributions automatically and enables the decision maker to compare directly with past NPV analyses.
As I have already explained, the problem associated with this technique is that it will double count the risk: firstly in the discount rate and then by representing the NPV as a distribution. However, if one is aware of this shortfall, the result is very useful in determining the probability of achieving the required discount rate (i.e. the probability of a positive NPV). The actual NPV to quote in a report would be the expected value of the NPV distribution.
Internal Rate of Return
The Internal Rate of Return (IRR) of a project is the discount rate applied to its future cashflows such that it produces a zero NPV. In other words, it is the discount rate that exactly balances the value of all costs and revenues of the project. If the cashflows are uncertain, the IRR will also be uncertain and therefore have a distribution associated with it.
A distribution of the possible IRRs is useful to determine the probability of achieving any specific discount rate and this can be compared with the probability other projects offer of achieving the target discount rate. It is not recommended that the distribution and associated statistics of possible IRRs be used for comparing projects because of the following properties of IRRs:
· Unlike the NPV calculation, there is no exact formula for calculating the IRR of a cashflow series. Instead, a first guess is usually required, from which the computer will make progressively more accurate estimates until it finds a value that produces an NPV as near to zero as required.
· If the cumulative cashflow position of the project passes through zero more than once, there is more than one valid solution to the IRR inequality. This is not normally a problem with deterministic models because the cumulative cashflow position can easily be monitored and the smallest of any IRR solutions selected. However, a risk analysis model is dynamic, making it difficult to appreciate its exact behaviour. Thus, the cumulative cashflow position may pass through zero and back in some of the risk analysis iterations and not be spotted. This can produce quite inaccurate distributions of possible IRRs. In order to avoid this problem, it may be worth including a couple of lines in your model that calculate the cumulative cashflow position and the number of times it passes through zero. If this is selected as a model output, you will be able to determine whether this is a statistically significant problem and alter the first guess to compensate for it.
· IRRs cannot be calculated for only positive or only negative cashflows. IRRs are therefore not useful for comparing between two purely negative or positive cashflow options, e.g. between hiring or buying a piece of equipment.
· It is difficult to compare distributions of IRR between two options unless the difference is very large. This is because a percentage increase in an IRR at low returns (e.g. from 3% to 4%) is of much greater real value than a percentage increase at high returns (e.g. from 30% to 31%). It is therefore very difficult to compare the value of two projects in terms of the IRR distributions they offer. One project may offer a long righthand tail that can easily increase the expected IRR but in real value terms this could easily be outweighed by a comparatively small diminishing of the lefthand tail of the other option.
An example of the nonlinear relationship between IRR and real (or present) value.
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