Project Cost with Risk

This is the final lesson of the first part of the course. Here we combine the results of previous lessons to obtain estimates of the capital cost for a project. The estimates explicitly consider risk. The WBS model is adapted to include probability distributions for cost components. Using formulas from probability theory we compute the mean and variance of the system costs. With these results we can compute several measures that assist decision-makers in risky circumstances.

For most situations the results obtained are approximate in that the normal distribution is used to model the capital cost. When this approximation is unacceptable, Monte-Carlo simulation is used to obtain results.

Because of time limitations, this lesson concentrates on total capital costs. The Estimate add-in uses the same methods to find moments when time is considered and net present worth is the measure of interest. The same methods, with some variations, are used to find moments for life-cycle costs.

 Goals
 Assume the task costs are modeled using independent normal distributions. Make point estimates and probability statements about the total capital cost. Assume the task costs are modeled using the uniform, triangular, beta or normal distribution. Assuming independence, find the mean and variance of the total capital cost. Approximating the distribution of the total cost with the normal distribution, make probability statements about the cost. Use the Estimate and Random Variables add-ins to define a project with non-normal distributions for task costs. Make point estimates of the total capital cost with mean values, percentiles and simulation. Use the Simulation RV feature of the Random Variables add-in to estimate capital cost and make probability statements without the normality assumption.
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The textbook does not include this topic.

 Capital Costs

We return to our assembly line example from the Capital Cost lesson (Lesson 3), but now we add probability distribution information to the model. The Estimate add-in has an option to construct models with random variables specified for task costs. The Random Variables add-in is used during the construction process, so it must be installed in Excel to define and analyze models with risk.

 Estimation with Risk

The data in the figure are only illustrative. The methods described here allow any continuous or discrete distribution included with the Random Variables add-in. The important assumption of this lesson and throughout this course, is that the random variables defined for the tasks must be statistically independent.

To continue, we must compute the mean and variance of each task cost. Although we can do this with the formulas provided in the Estimation with Risk lesson, the Random Variables add-in computes them automatically. Three columns to the right of the distribution information display the mean, variance and standard deviation of the each cost. Click the icon below to see the larger area of the worksheet.

 WBS with Triangular Distributions

The arrows in the picture show the Excel functions that compute the mean and variance. The standard deviation is the square root of the variance.

The worksheet also provides a point estimate for each cost, here in column L. We have chosen the distribution mean for the point estimate. Because each distribution is skewed to the right, the mean is greater than the most likely value. The percentile or a simulated value could also have been selected. These options are discussed below.

 Distribution of the Capital Cost

The total capital cost is the sum of the task costs.

We assume that the task costs are independent random variables. Then we can use two results from probability theory to derive the moments of the capital cost.

 The mean of the sum of random variables is equal to the sum of the means of the random variables.

 The variance of the sum of independent random variables is equal to the sum of the variances of the random variables.

For our model, we compute the mean and variance of the capital cost by the following equations.

The summary provided by the add-in performs these computations and reports the moments for each level 2 task and the total capital cost.

 WBS Summary

The point estimate of the capital cost of the project is at the extreme right of the figure. This is simply the sum of the mean value estimates, or \$328,533. Recall that our earlier analysis (in the Capital Cost lesson) estimated the capital cost as \$308,000. This estimate summed the most likely values of the triangular distributions. Because these distributions are skewed to the right, the sum of the mean values is greater than the sum of the most likely values.

 Probability Statements

In order to continue the analysis we will make an approximating assumption based on the Central Limit Theorem of probability theory.

 The Central Limit Theorem states that the sum of many independent, identically distributed random variables is approximately normally distributed.

When the summed random variables have normal distributions a stronger result applies.

 The sum of independent, normally distributed random variables is normally distributed.

For practical instances, it will rarely be true that the conditions in these two boxes are satisfied. Nevertheless, we will find it convenient to make probability statements about the project cost using the normal distribution. When this assumption is not warranted we use simulation as discussed below. In the statement below, C is the the random variable that represents the capital cost for the project.

This formula may be used, for example, to compute the probability of getting a lower value than the value determined assuming the most-likely cost for all tasks. Using the standard normal distribution we calculate that there is only about a 11% chance that the capital cost will be that low.

Textbooks often provide tables for the approximate evaluation of the standard normal distribution. The link following shows such a table. The result just calculated can be found approximately from this table for z = -1.25 instead of -1.235.

 Standard Normal Cumulative

When working in Excel it is not necessary to use the standard normal distribution for the calculations. The figure below shows two approaches for the example. The one on the left uses the built-in NORMDIST function. The one on the right uses the Random Variables add-in function RV_prob.

 Interval Estimate

When we know its distribution, we can make interval statements about the project cost. The interval estimate is a range of the random variable that covers a given probability. The equations below describe a two-sided confidence interval with equal probabilities on both sides of the range. One-sided intervals compute one or the other limit without dividing the confidence level by 2. The other limit is unbounded.

When the project cost is normally distributed, the interval is found with the inverse normal distribution function. When the project cost is approximated by a normal distribution or when the probabilities are determined by simulation, the coverage level should not be too high. The approximation of the tails of the distribution may be poor and the interval may not be accurate. The equations below use the inverse of the standard normal.

For the example, we will now find the interval with 0.9 coverage. Given the estimates and assuming normality, there is a 0.9 probability that the project cost will fall within this range; that is,

These results were computed with Excel functions. Approximate results can be found with the standard normal inverse table.

 Standard Normal Inverse

Excel provides a function called NORMINV to compute the inverse of the normal distribution. The bounds on the left in the following figure are computed with this function. The Random Variables add-in uses the function RV_inverse to compute inverse probabilities as shown at the right.

 Percentile Point Estimate

We earlier used the mean values of the task estimates to estimate the project costs, but there are other possibilities. In the last lesson, Estimation with Risk, we mentioned the mode, median, percentile, and simulation as ways to make point estimates. The Estimate add-in allows the mean, percentile, and simulation methods. The median is the 50th percentile. To choose a different method, click the Change button at the top of the Estimate worksheet. We consider here using the percentile.

Say all the persons estimating the task costs are conservative. They estimate costs that cannot be exceeded more than 40% of the time. That is, they estimate the task costs at the 60th percentile. The Random Variables add-in computes percentiles with the inverse probability function. Click the icon to see the results for the example problem.

 Percentile Point Estimates

The figure shows that with the normal distribution assumption for the project cost, the probability that this cost is less than or equal to the sum of the estimates is 66%. Although the individual task estimates are at the 60th percentile, the project estimate is at the 66th percentile. The sum of conservative estimates results in an even more conservative project cost estimate.

The actual project cost, when realized, is the sum of independent random variables. Some will be greater than the mean and some will be less than the mean. There is a 66% chance that this sum will be less than the value obtained as the sum of the 60th percentile estimates.

 Simulation

When the task costs are not normally distributed it is not accurate to assume the summed task costs are normally distributed. Although, in many cases the normality assumption may yield fairly accurate answers, the simulation approach does not require normality. Simulation is a very powerful tool that can be used in many contexts. Here we apply it to estimate the project cost.

First, click the Change button to make simulation the estimation method for task costs. Each point estimate on the worksheet will be replaced by a function that simulates a value every time the worksheet is recalculated. This occurs automatically when a data item is changed and also when the user initiates a recalculation command. For the Windows OS, the "Ctrl-=" causes recalculation. For the Mac OS, the "Command-=" causes recalculation. Click the icon below to see the worksheet with simulated values.

 Simulation Point Estimates

Column L holds the simulated values. The sum of the values is just below the column in L25. The same value is included the summary in cell X14. Excel changes all the simulated values every time the worksheet is recalculated. The sum that represents the project cost changes with each recalculation. We have gathered five observations in column X with the mean and standard deviation of these five numbers immediately below. The five numbers show the variability of simulated results. The variability represents the uncertainty in the project cost estimate.

So what information does this provide? One simulated value or even five gives no meaningful information. By simulating a large number of observations, however, one learns a great deal about the distribution of the project cost or any function of the simulated values. A single simulated value is one realization of the population of all project costs. To learn more about the population we use the Random Variables add-in to simulate 1000 observations.

The window that opens when you click on the next icon explains how to use the Estimate and Random Variables add-ins for simulation. The text describes several steps that become visible when you scroll down the page.

 Simulation Summary

The figure below shows the results of a sample of 1000 simulated observations. The numbers are slightly different than obtained in the summary because they are from a different simulation run. The bottom three numbers in column Z are statistics computed from the 1000 observations. The simulated mean and standard deviation are nearly those computed from probability theory. The skewness is positive indicating that the distribution of project cost is not symmetric as it would be if the project cost were actually normally distributed. Since our assumed distributions have a positive skew, this is expected.

The Random Variables add-in also provides a histogram of the distribution of the observations. It is skewed to the right.

Based on the statistics collected during the simulation, the add-in constructs a new random variable, named SimX14 for the example. The name comes from the location of the simulated cell on the worksheet. Using this distribution, the RV_prob and RV_inverse functions can be used to obtain the probability and interval estimates described earlier. In contrast to the earlier calculations, the results below do not depend on the normality assumption for the project cost. They are obtained by repeatedly simulating the task costs and collecting data on the project cost. The next figure gives the estimates that were obtained from the histogram constructed from the simulation runs. The results themselves are random variables. If the system were simulated again, the estimates that we see here would change.

Simulation is a valuable analytical tool. It is used in almost every field in which uncertainty is modeled with random variables.

 Summary
 Project Cost with Risk Summary

Engineering Finance
by Paul A. Jensen