1. Identifying Input Distributions 1. Fit Distribution to Historical Data 2. 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

2. 1. Using Observed Data to Fit Distributions Group data into histograms or cumulative probability distributions Assume a distribution shape and estimate its parameters Adjust the extreme values if appropriate Perform Goodness of Fit Tests to see if distribution could produce observed data: ◦ Chi-Square Test, Kolmogorov-Smirnoff (K-S) Stat ◦ Overlay graphs

3. 2. Assuming Distributions Example: Modeling the Price of a Stock Most 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: P t = P 0 * exp[( μ -.5* σ 2 )*t + σ *Z*t.5 ] where the parameters μ and σ are based on historical numbers or market research

4. Regression Forecast Models In a linear regression model, Y= b 0 + b 1 X + e ◦ b 0 = the y intercept of the line ◦ b 1 = 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

5. Using Regression Models to Forecast Distributions Regression 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.

6. Excel’s Dialog Box for the Excel Sample Data

7. Linear Trendline Forecasts: the 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 b 1 represents the expected growth (in $1,000s) during one period (year) Here: b 0 + b 1 X is the forecast for sales in year X

8. Output Interpretation R 2 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.

9. Distribution for Base Value Forecast Y will be ◦ Normally distributed with ◦ μ = b 0 + b 1 X ◦ σ = model’s standard error (SE of the regression) Excel formula for Year 6 sales: ◦ =norminv(rand(),61.248,2.65)

10. Sales Growth Rate% g: the Compound Growth Model Forecast Sales t = Sales 0 (1+ g) t ◦ ln(Sales t ) = b 0 + b 1 t where b 1 = ln(1+g) ◦ Therefore g = e b1 - 1 ◦ The sales growth rate g will be Normally distributed with ◦ μ = e b1 - 1 ◦ σ = e b1 standard error - 1 Excel formula for year 6 sales growth rate % ◦ =norminv(rand(),.2344,.0077)

11. Forecasting % of Sales Distributions Forecast Total Assets as a percent of sales. Using a linear regression model, ◦ Y=Total Assets = b 1 X ◦ X= Sales ◦ Constant set = 0 ◦ b 1 = percent of sales estimate b 1 will be normally distributed with ◦ μ = b 1 = X variable 1 coefficient in Excel ◦ σ = b 1 standard error = X variable 1 standard error in Excel ◦ =norminv(rand(),.686,.0625)

12. 3. Use Experts: Common 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 should Adjustment and Anchoring Unwillingness to consider extremes Organization culture & conflicting agendas Estimation units are unfamiliar

13. Modeling Techniques to Elicit Expert Opinions Disaggregation of Input Random Variables Brainstorming Sessions & Individual Follow-up Choice 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.