Identifying Input Identifying Input DistributionsDistributions1. Fit Distribution to Historical Data2. Forecast Future Performance

and Uncertainty◦ Assume Distribution Shape and

Forecast Parameter Values Based on Historical Data

3. Solicit Expert Opinions when Data is not Available

1. 1. Using Observed Data to Fit Using Observed Data to Fit DistributionsDistributionsGroup data into histograms or cumulative

probability distributionsAssume a distribution shape and estimate

its parametersAdjust the extreme values if appropriatePerform Goodness of Fit Tests to see if

distribution could produce observed data:◦ Chi-Square Test, Kolmogorov-Smirnoff (K-S)

Stat◦ Overlay graphs

2. 2. Assuming DistributionsAssuming Distributions Example: Modeling the Price of Example: Modeling the Price of

a Stocka StockMost financial models of stock prices

assume that the stock’s price follows a lognormal distribution. (The logarithm of the stock’s price is normally distributed so its returns are normally distributed)

This implies the following relationship: Pt = P0 * exp[(μ-.5*σ2)*t + σ*Z*t.5]

where the parameters μ and σ are based on historical numbers or market research

Regression Forecast Regression Forecast ModelsModelsIn a linear regression model,

Y= b0 + b1 X + e

◦b0 = the y intercept of the line

◦b1 = the slope of the line which is a measure of growth per unit change in X

◦X = the time period or dependent variable being used to predict Y

◦e = random error term

◦Y = the variable being forecasted

Using Regression Models to Using Regression Models to Forecast DistributionsForecast DistributionsRegression Theory states that

forecasted numbers are expected to be Normally distributed with an Expected Value equal to the model’s predicted value and a Standard Deviation equal to a function of the model’s standard error.

Regression is done in Excel using the Tools Data analysis Regression menu option.

Excel’s Dialog Box for the Excel’s Dialog Box for the Excel Sample DataExcel Sample Data

Linear Trendline Forecasts: Linear Trendline Forecasts: the Constant Change Modelthe Constant Change Model

Y is the dependent variable being forecasted (such as sales in $1,000s in column B)

X is the independent variable that is a measure of time (such as the year in column A) and that is being used to explain the dependent variable

b1 represents the expected growth (in $1,000s) during one period (year)

Here: b0 + b1 X is the forecast for sales in year X

Output InterpretationOutput InterpretationR2 is the percent of variation in Y that is

explained by the regression model used on X. It will be a number between 0 and 1, where 0 represents none of the variation being explained and 1 represents 100% of the variation being explained.

The standard error of the model is the average amount of scatter around the predicted forecast line. It describes how far actual values have fallen from the line on average.

Distribution for Base Value Distribution for Base Value ForecastForecastY will be

◦Normally distributed with◦μ = b0 + b1 X

◦σ = model’s standard error (SE of the regression)

Excel formula for Year 6 sales:◦=norminv(rand(),61.248,2.65)

Sales Growth Rate% Sales Growth Rate% gg: : the Compound Growth Modelthe Compound Growth ModelForecast Salest = Sales0 (1+ g)t

◦ ln(Salest ) = b0 + b1 t where b1 = ln(1+g)

◦ Therefore g = eb1 - 1

◦The sales growth rate g will be Normally distributed with

◦ μ = eb1 - 1◦ σ = eb1 standard error - 1

Excel formula for year 6 sales growth rate %◦ =norminv(rand(),.2344,.0077)

Forecasting % of Sales Forecasting % of Sales DistributionsDistributionsForecast Total Assets as a percent of

sales. Using a linear regression model,◦ Y=Total Assets = b1 X ◦ X= Sales◦ Constant set = 0◦ b1= percent of sales estimate

b1 will be normally distributed with ◦ μ = b1 = X variable 1 coefficient in Excel◦ σ = b1 standard error = X variable 1

standard error in Excel◦ =norminv(rand(),.686,.0625)

3. 3. Use Experts:Use Experts: Common Biases and ErrorsCommon Biases and Errors

Perception limited to information and experiences ◦ Bias of most likely value◦ Wider ranges of uncertainty

Inexpert expert◦ Know it all who prescribes narrower ranges

of uncertainty than shouldAdjustment and AnchoringUnwillingness to consider extremesOrganization culture & conflicting

agendasEstimation units are unfamiliar

Modeling Techniques to Elicit Expert Modeling Techniques to Elicit Expert OpinionsOpinionsDisaggregation of Input Random VariablesBrainstorming Sessions & Individual Follow-

upChoice of Distribution Encouraged:

nonparametric is preferred (uniform, triangular, betapert)

In eliciting 3 point values, give worst case scenario first to get minimum estimate, best case next for maximum and then ask for most likely estimate

Give visual aids such as histograms to ask questions about likelihoods.