Engineering Finance

Estimation with Risk

This lesson adds measures of risk to the estimation process. Uncertainty of estimation is always present and in our view, it is better to account for it explicitly rather than to simply ignore it. The methods for dealing with risk introduced here will be used throughout the course.

The textbook describes a variety of types of risks associated with projects, but our analytic methods concentrate on the uncertainties inherent in the estimates computed by the WBS and CBS models. This lesson uses probability distributions to model the uncertainties of cost and revenue estimates. In the next lesson, the distributions are combined to produce moments for the capital cost model. Both the Random Variables add-in and several tables are used for the computations.

The lesson assumes that students have some background in probability and statistics. Students should be able to compute simple statistics such as the mean and standard deviation of data and be sufficiently familiar with the continuous distributions used in this section to accomplish the goals listed below.

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  • List the sources of uncertainty associated with a project.
  • Use probability distributions to make estimates that explicitly measure risk.
  • Be familiar with four distributions used in this lesson: Uniform, Triangular, Beta and Normal. For each case, know how the distribution parameters affect the shape of the distribution.
  • Use the Random Variables add-in to compute for a given distribution the mean and standard deviation. Be able to evaluate probability and inverse probability statements. Be able to simulate random variables.
  • Use the Random Variables add-in to make point estimates using the expected value, the kth percentile and simulation.
  • For the normal distribution, use tables to make point estimates using the expected, the kth percentile and simulation. Also be able to evaluate probability and inverse probability statements.

There are many sources of risk and uncertainty in engineering projects. The referenced sections in the text provide some general remarks regarding the source and management of risk. Be sure to read them to find the variety of reasons why the success of a project is uncertain and ways to reduce or respond to risk during project planning. The lesson concentrates on uncertainties in costs or revenues, but students should be aware of the other sources as well.


1.3.2 Risk and Uncertainty


8.4.4 Risk Management



Probability Distributions

We use continuous probability for estimates that admit uncertainty. With probability distributions, we have a mathematical basis for analyzing the effects of decisions on risk. Probability theory and simulation procedures provide many useful tools for the analyst and decision maker.

With a probability distribution representing a cost, the actual cost that will be observed is a random variable, indicated as x below. For this discussion we use continuous random variables. The discussion refers to cost, but revenue items are similarly estimated.

We also use the inverse cumulative function.

Although we focus on four named distributions, the Random Variables add-in allows a variety of named continuous and discrete distributions. The link below opens a page in the ORMM site that lists the distributions available.


Several continuous probability distributions, specifically identified by the Estimate add-in, are listed below. Others also may be used.

Range Estimate

Here the estimator provides lower and upper bounds for the cost. The lower bound is a and the upper bound is b. The range specification provides no information about the distribution of values within the range so we use the uniform distribution. This assumes all values are equally likely.

Uniform Distribution

Three-Point Estimate

A traditional way to get subjective estimates from an expert is to ask three questions: what is the smallest you think the cost can be? what is the most likely value? what is the largest value? The triangular distribution is defined exactly by these three measures identified respectively by the letters, a for the smallest, m for the most likely, and b for the largest. The most likely value for a distribution is called the mode.

Triangular Distribution

Beta Estimate

This is an upgrade to both the range and three-point estimates. The Beta distribution is defined by its lower and upper bounds, a for the lower bound and b for the upper bound, as well as two parameters defining its shape and . A variety of shapes can be defined. The mode is a function of the parameters.

Beta Distribution

Normal Estimate

The normal distribution has the traditional bell-shape appropriate for many measurements. The estimate is defined by two parameters the mean, , and the standard deviation, . When we estimate a cost based on the average of a large number of observations. The statistical mean and standard deviation provide estimates of the distribution parameters.

Normal Distribution

Other Distributions

The Random Variables add-in provides a variety of other distributions that might be useful in modeling the variability of estimates.

Other Distributions

Using a probability distribution to represent the uncertainty of an estimate complicates the estimation problem because rather than a single point estimate, there are two or more estimates necessary for the parameters of the distribution. The additional effort is well rewarded by the many useful results that can be obtained from probability theory.

Information from Distributions

Once a distribution is specified with its parameters, several useful quantities may be computed using mathematical formulas or numerical methods. This course uses the Random Variables add-in. The add-in uses numerical methods for computations. For this lesson, the student should be able to define random variables and use the Excel functions listed in the table below to discover information about the distribution.

Random Variables Add-in

The figure below shows a random variable representing the cost of a work package in a WBS. The definition of the random variable, called WP1, is placed on the worksheet by the add-in. The definition has been placed in the range of cells D5:D9. The definition consists of the name of the distribution (Beta) and its parameters, alpha, beta, lower and upper. The cells below the definition hold functions that are provided by the add-in to compute information about the distribution. Although in this case the functions are placed below the definition, they can be placed anywhere on the worksheet. Be aware that the notation of the add-in uses e rather than w in the figure.

The quantities computed should be familiar to a student with a background in probability theory. They are described in the following table with their Excel function. These functions are available only when the Random Variable add-in is installed.


Excel Function

Mean Value
The expected value of the random variable. This is often used as a point estimate.
Represents the variability of the random variable. Another useful quantity is the standard deviation that is the square root of the variance.

The probability that the random variable falls within the range c to d. If c left blank, the lower limit is negative infinity. If d left blank, the upper limit is positive infinity.

P(c < x < d)


Inverse Probability

Computes the value of the random variable such that the probability that the random variable is less than that value is equal to w.

The median is RV_INVERSE(WP1,0.5)

The 90th percentile is RV_INVERSE(WP1,0.9)

Simulates the random variable. The function name ending in V indicates that this function is volatile. Every time the worksheet is computed a new value is simulated. This is useful for Monte-Carlo simulation.

In the next lesson we learn how these functions are used to compute project cost estimates.


There are many situations that are affected by more than one random variable.  For all but the simplest, probability models will be too difficult to construct and evaluate.  Rather, it is necessary to simulate the several random variables that describe the situation and observe the results. By running a simulation model many times, statistical analysis provides the estimates we need. In this lesson we describe the reverse transformation method for simulating a single random variable. In the next lesson we use simulation to estimate the capital costs of a project.

To simulate a random variable with a known distribution, choose a random number from a uniform distribution bounded between zero and one. Find the value of the random variable for which the cumulative distribution is equal to the random number. This is the same as finding the inverse cumulative value. Report that value as the simulated value. Mathematically, this is the process shown below.

The process is illustrated below for a normal distribution with mean 10 and standard deviation 3. The figure shows the cumulative distribution. To simulate an observation, choose a random number. For the illustration we choose r as 0.73. Set that number to the value of F(x) on the vertical axis. Project over to the cumulative distribution and then down to the x-axis to find the simulated value. For r equal to 0.73, the value of x is 11.83.

For a few simple distributions an explicit form of the inverse cumulative is available, and simulation can be done analytically. For those that cannot numerical methods are used. TheRandom Variables add-in uses the RV_SIMV(WP1) function. The argument is the name of a distribution defined on the worksheet by the add-in.

There are many sources for random numbers that may be used for simulation. Excel has a random number generator. The link below opens a table of random variables generated by Excel.

Random Numbers
Normal Distribution

To illustrate the methods of this lesson, it is helpful to consider the normal distribution. We use this distribution to describe capital cost estimates. We provide tables below that are useful for these purposes. The tables are constructed for Standard Normal distribution with mean 0 and standard deviation 1. For the more general normal variant we must transform the results.

For solving problems regarding the normal distribution we provide tables of the cumulative and inverse cumulative functions for the Standard Normal. Click on the icons to see the tables.

Standard Normal Cumulative


Standard Normal Inverse


Point Estimate

Say you have chosen a distribution. Your boss doesn't really understand this newfangled concept of probability and he proclaims, "Give me a number. Just give me a number." He wants a point estimate, so what should you tell him?

The most familiar, and often the most reasonable response is the expected value or the mean of the distribution. Another possibility is the median or the middle value. Another measure of central tendency is the mode. The mode is the value with the highest value of the density function. If you are feeling conservative and want to estimate a value greater than the middle value you might give him the kth percentile with k greater than 50%. If you think he's looking for a low cost estimate you might give him an aggressive estimate as the kth percentile with k less than 50%. The formulas below describe the mean, median and percentile estimates using probability notation. We use x' as the value of the point estimate.

We could always simulate a value using the Monte-Carlo method and report that value as the point estimate. Your boss might find it disconcerting, however, that you keep changing your estimate every time he asks, but one simulated value is as good as any other. Perhaps giving your boss a point estimate that is simulated is not a good policy, but simulation is very useful for estimating a quantity that involves several random variables.

For some probability distributions the point estimates can be determined analytically. The linked document has the available formulas for the four distributions of this page.

Point Estimate Formulas

When analytical estimates are inconvenient or impossible, the numerical procedures of the Random Variables add-in provide all of these point estimates except the mode.

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Estimation with Risk Summary


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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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