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Internal Rate of Return If we have a set of expected cash inflows and cash outflows, the internal rate of return is the discount rate that equates the present values of the two sets of flows. If A, is an expected cash outflow in the period i and Rt is the expected inflow for the period t, the internal rate of return is the value of k that satisfies the following equation (note that the A0 will be positive in this formulation of the problem):

A0 + A,/(l + k) + A2/(l + kf + . . . + AAI + k)n = /?,/(! + k) + R2/{ 1 + kf + . . . + + fc)" t = 1,2,3 it

### The value of k is found by trial and error.

Profitability Index Also known as the benefit-cost ratio, the profitability index is the net present value of all future expected cash flows divided by the initial cash investment. (Some firms do not discount the cash flows in making this calculation.) If this ratio is greater than 1.0, the project may be accepted.

Other Prolitability Models There are a great many variations of the models just described. These variations fall into three general categories: (1) those that subdivide net cash flow into the elements that comprise the net flow, (2) those that include specific terms to introduce risk (or uncertainty, which is treated as risk) into the evaluation, and (3) those that extend the analysis to consider effects that the project might have on other projects or activities in the organization. Two product line extension models, taken from Dean 116), will illustrate these methods.

A project with this payback period would probably be considered quite desirable.

Pacifico's Method PI is the profitability index of acceptability where PI = rdpc SP VL/C, r = probability of research success, d = probability of development success, given research success, p = probability of process success, given development success, and c = probability of commercial success, given process success.

The investment, C, is the estimated total cost of the R & D effort for the project. Risk is incorporated in the rdpc term. The cash flow is SP VZ where

S — estimated average annual sales volume in units of product, P = estimated average annual profit per unit, and

L = estimated life of the product extension in years. (Note that although the profits are not formally discounted, they are "devalued" over time by multiplying them by VZ rather than by L.)

Dean's Profitability Method Dean's model contains a term that subtracts the unit manufacturing cost and the unit selling and administrative costs from the unit price, multiplies the remainder by the expected number of units sold per year, and then subtracts tooling and development costs (a project risk factor is also included). All costs and revenues are time-indexed and discounted to the present. Dean modifies his model to deal with three distinct cases: (1) where the product extension has no significant impact on the existing system, (2) where the product extension may affect the profitability or the sales of existing products, or both, and (3) where the product extension is a replacement for an existing product.

Several comments are in order about all the profit-profitability numeric models. First, let us consider their advantages.

1. The undiscounted models are simple to use and understand.

2. All use readily available accounting data to determine the cash flows.

3. Model output is in terms familiar to business decision makers.

4. With a few exceptions, model output is on an "absolute" profit/profitability scale and allows "absolute" go/no-go decisions.

5. Some profit models account for project risk.

6. Dean's model includes the impact of the project on the rest of the organization. The disadvantages of these models are the following.

1. These models ignore all nonmonetary factors except risk.

2. Models that do not include discounting ignore the timing of the cash flows and the time value of money.

3. Models that reduce cash flows to their present value are strongly biased toward the short run.

4. Payback-type models ignore cash flows beyond the payback period.

5. The IRR model can result in multiple solutions.

6. All are sensitive to errors in the input data for the early years of the project.

7. All discounting models are nonlinear, and the effects of changes (or errors) in the variables or parameters are generally not obvious to most decision makers.

8. Those models incorporating the risks of research and/or development and/or process (the commercial success risk factor is excluded from this comment) mislead the decision maker. It is not so much that the research-development-process success is risky as it is that the time and cost required to ensure project success is uncertain. The application of these risk terms applies mainly to R & D projects.

9. Some models, Dean's and Pacifico's, for example, are oriented only toward evaluation of projects that result in new products.

10. All these models depend for input on a determination of cash flows, but it is not clear exactly how the concept of cash flow is properly defined for the purpose of evaluating projects. (This problem is discussed later in this chapter.)

A complete discussion of profit/profitability models can be found in any standard work on financial management—see 11, 9, 67], for example. In general, the net present value models are preferred to the internal rate of return models.

In our experience the payback period model, occasionally using discounted cash flows, is one of the most commonly used models for evaluating projects and other investment opportunities. Managers generally feel that insistence on short payout periods tends to minimize the uncertainties associated with the passage of time. While this is certainly logical, we prefer evaluation methods that discount cash flows and deal with uncertainty more directly by considering specific risks. Using the payout period as a cash-budgeting tool aside, its only virtue is simplicity, a dubious virtue at best.

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In the early 1980s, the U.S. Air Force found that it needed to be able to predict the flight test costs (one of eight cost predictions required for a full estimate) of electronic warfare systems such as radar warning receivers, electronic countermeasure radiating devices, and chaff dispensers. This task was an exceptionally difficult one due not only to the rapid technological advancements being made in the electronics field, but particularly because the estimates had to be made up to six years before the equipment would incur those costs; that is, even before the system was conceptualized or defined!

Sour«: |.R. Ward, "Project Management Cost Estimate: A Case Study in Electronic Warfare System Flight Test Costs," Project Management Iournal, December 1984.

Cost estimating approaches based on estimating the costs of the system components could not be used because the components were often not even identified six years beforehand. Sometimes the only data available was the nature of the enemy system to be countered, or possibly, only the general type of equipment to be used. Thus, an approach was used based on independent variables with strong causal links to the dependent variable of interest: the flight test cost. Two general rules were employed to select variables to be included in a variety of statistical models available for test: (1) a logical, causal relationship must exist between the variable and the flight test cost, and (2) the variable must relate to an equipment characteristic that can be identified early in the equipment's conceptual phase. Based on these rules, the following variables were selected for testing:

weight, density, volume, input power, equipment type, whether new or modified equipment, and the phase of development for the flight tests.

Two statistical models offered good results (high correlations with actual flight test costs) but had problems unique to each of them. A linear regression model was unable to employ a number of key variables that would have offered good predictive ability. A principle components analysis model had two problems: (1) it used many cross-product variables whose causality was unclear, and (2) it only allowed a point estimate to be made without a confidence interval or trade-off figures.

Nevertheless, the study was considered highly successful, offering almost 90 percent explained variance for systems six years prior to actual use.

## Project Management Made Easy

What you need to know about… Project Management Made Easy! Project management consists of more than just a large building project and can encompass small projects as well. No matter what the size of your project, you need to have some sort of project management. How you manage your project has everything to do with its outcome.

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