Download Identifying Input Distributions

Identifying Input Distributions
Fit Distribution to Historical Data
Forecast Future Performance and
Uncertainty
1.
2.
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3.
Assume Distribution Shape and Forecast
Parameter Values Based on Historical Data
Solicit Expert Opinions when Data is
not Available
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
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:
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 Models
In 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
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.

Excel’s Dialog Box for the Excel
Sample Data
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
 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 Interpretation
R2 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 Forecast

Y 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% g:
the Compound Growth Model

Forecast 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 Distributions

Forecast 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. 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

Modeling Techniques to Elicit Expert Opinions
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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.