Evaluation with Risk

The previous lessons on evaluating and comparing projects have all neglected uncertainty. We have computed the net present worth, net annual worth and rate of return using single values for the parameters that define the cash flow. An example is shown below. It has a single investment of \$1,800 initially, and the revenue is uniform at \$650 per year for the 5 years of the project life. The operating cost begins at \$100 in the first year and increases as an arithmetic gradient of \$50 per year. Finally at the end of the 5 years there is a salvage value of \$400. The evaluation form constructed by the Economics add-in is shown below. With an MARR of 10%, the NPW and NAW are positive so the investment is acceptable. In fact, the IRR of the project is almost 14%.

As with all economic evaluations the numbers used in the model are only estimates. In this example, the magnitudes of the cash flow components indicated in cells C10 through C14 are really not known with certainty. The life of the project is also uncertain. Generally, we suspect that estimates of expenditures and revenues early in the project are better than those that occur later. How should we deal with this uncertainty in the decision making process?

A variety of ways have been proposed to adjust the numbers in the deterministic analysis to reflect risk. One popular method is to increase the MARR for risky projects. Then profits far in the future will contribute less to the NPW. Alternatively, by arbitrarily reducing the planned life of the project, profits beyond the reduced life are entirely neglected. Sensitivity and breakeven analysis are methods that analyze the variability of the parameters one at a time.

In this lesson we explicitly recognize risk by assigning probability distributions to uncertain estimates. We then use probability theory and Monte Carlo simulation to draw conclusions about the statistical variability of the evaluation measures.

 Goals
 For a given cash flow, compute the sensitivity and the breakeven point for any single parameter of the cash flow. Construct a spider chart showing sensitivity to several parameters. For a cash flow defined by a set of cash flow components with uncertain values, find the mean and variance of the NPW and NAW. Assuming normally distributed and independent random variables make probability statements regarding the NPW and NAW. Use the Economics and Random Variables add-ins for assigning probability distributions to cost estimates and to estimate the mean and variance of the NPW and NAW. Use the Economics and Random Variables add-ins to perform Monte-Carlo simulation to estimate statistics and probabilities concerning NPW and NAW.
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 3.5 SENSITIVITY AND BREAKEVEN ANALYSIS
 NPW Function

The NPW of the example is computed by multiplying the estimated values of each cash flow component by the appropriate equivalency factor and summing. For an MARR of 10% we have

The models of this section allow the estimates to change, so it is useful to have a more general expression for NPW. In the following expression, is the equivalency factor for term k. This factor is a constant for given values of the MARR and the project life. The quantity is the cash flow estimate for term k.

When the life is held fixed, this expression is easy to evaluate because all the values are constant and the NPW value is a linear function of the estimates. For simple analyses, we will often make this assumption. When the life is allowed to vary, however, the function is nonlinear and for most problems the function is not easily written as a function of the project life N. The Economics add-in can easily evaluate the NPW and NAW for given values of N, allowing numerical analysis of variations in the estimate of life.

In our example, there are five terms in this expression. The first is for the investment, the second for the annual revenue and so on. The table shows the estimates, factors, the contribution to the NPW of each component and the resultant sum that is equal to the project NPW.

 Index Component Estimate (x) Equivalency factor Factor value (C) Cx 1 Investment -1800 1 1 -1800 2 Revenue 650 (P/A, 0.1, 5) 3.7908 2464 3 Op. Cost -100 (P/A, 0.1, 5) 3.7908 -379 4 Op. Gradient -50 (P/G, 0.1, 5) 6.8618 -343 5 Salvage 400 (P/F, 0.1, 5) 0.6209 248 Sum Cx to find the NPW = 190

Given the life of the project the NAW is similarly calculated.

The expressions show that the measures of economic acceptability, NPW and NAW, are functions of the cash flow component estimates, the interest rate used for the evaluation, and the estimate of the life. In the following we illustrate approaches to the decision problem when the uncertainty of the estimates is explicitly considered.

 Sensitivity Analysis

The purpose of sensitivity analysis is to show how changes in the individual estimates affect the economic measures and the decision to accept or reject a project. The method selects a base value for each estimated value. Then each estimate is individually allowed to change above and below the base value while holding all other estimates at their base values. The expression below shows how the NPW varies as a function of the variations from the base values.

The formula shows that the NPW is equal to a constant plus a linear expression that involves the variations. The constant term is the NPW we computed with the original estimates and the coefficients of the linear terms are the contributions of each component to the base net present worth. For the example the function is:

For sensitivity analysis, we only allow one component to change at a time, holding all other variations at zero, and observe the variation in the NPW.

The figure below, called a spider chart, is useful for displaying sensitivity results. Each line shows the variation of the NPW with one of the cash flow parameters. All lines cross at the base NPW value of 190. As expected the lines are straight. The slope of a line indicates the sensitivity of the NPW value to changes in a parameters. The red line representing revenue has the greatest slope, so the NPW is most sensitive to changes in revenue. It is least sensitive to changes in the salvage value.

Of particular interest is the portion of the line that falls below the NPW = 0 line. For these values of the parameter being changed, the return on the project is less than the MARR.

 Breakeven Analysis

For breakeven analysis, we hold all but one of the variations to zero and compute the value of remaining variation that makes the NPW equal to zero. For the cash flow component values, the linear equations are easy to solve.

The table shows the breakeven variations and the associated breakeven values that cause the NPW to be zero. The columns show the factors in the equation above. If the breakeven values are far from the values that might reasonably be expected, the decision maker might be confident in a decision to accept the investment. Alternatively, a value that is close to the base case might suggest that more study should be devoted to finding a more accurate estimate.

 Component Base (b) Factor (C) Variation () Value () Investment -1800 1.0000 0.106 -1990 Revenue 650 3.7908 -0.077 600 Operating cost -100 3.7908 0.502 -150 Operating gradient -50 6.8618 0.554 -78 Salvage 400 0.6209 -0.766 94
 Project Life

The sensitivity analysis for project life is more complicated because the NPW is a nonlinear function of the life of the project as reflected in the formula for our example. Also our model is only valid for integer values of the life. We include the expression for NAW because it is not meaningful to compare values of the NPW over different lives.

The chart shows the variation of NAW with life. It is clearly nonlinear. The breakeven point is at about four years.

 Mean and Variance of NPW

In this section, we model the amounts of each component as independent random variables. The equivalency factors for fixed values of MARR and project life are constants.

When the component values are independent random variables, the mean and variance of the NPW can be written as sums of terms. We use F as an indicator of the distribution of the values, because these results are not restricted to normal distributions.

The linked document shows the computation of the mean and variance of the example when the moments of the cash flow component values are given.

 NPW Example

With the assumed data we find the parameters of the NPW distribution.

The net annual worth, NAW, is proportional to the NPW so we can easily compute the moments of the NAW.

When we further assume that the component values are normally distributed, the NPW and NAW also have normal distributions. We then can use the Normal tables to find probability values. A particularly useful value is the probability that the NPW is less than zero. This is the probability that the project does not earn the MARR.

So for the example, there is about a 21% chance that the project will fail to meet the 10% MARR requirement.

This section has demonstrated two important results.

 When the cash flow components are independent random variables, the mean and variance of the NPW and NAW can be computed by weighted sums of the means and variances of the component distributions.

 When the cash flow components are independent normally distributed random variables, the NPW and NAW are normally distributed.

On the next page we use simulation to evaluate systems when the variability is not normal and when the life is allowed to be a random variable.

Engineering Finance
by Paul A. Jensen