Engineering Finance

Solution Alternatives

This is a review of the solution alternatives section of the course. This review cannot teach all the materials of the section. A careful student will at least skim the contents of each of the individual lessons focusing on concepts and techniques not immediately clear. If extra problems are provided, the student should try them and check against the answer keys.

It would also be useful for review to visit the presentations list in the resources section. Here the lecture presentations are listed for easy viewing. Use the calculators in the Toolbox to solve the example problems.

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Each lesson in the section has a collection of goals. Most of them describe an activity that you should understand and be able to perform. The paragraphs below describe the purpose of each lesson. Click the bush icon to see the goals of the individual lessons.

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  • Present and Annual Worth
    This lesson uses the present and annual worth criteria for selection between mutually exclusive alternatives. Both methods yield the same selection, but we will see that there are important differences in application and interpretation.
  • Rate of Return
    This lesson explores the rate of return method for comparing alternatives. We learn the important lesson that the rate of return method must be applied to increments of investment rather than directly to the individual alternatives.
  • Inflation
    Inflation of costs and revenues is inevitable in modern economies and it complicates economic decision making. Although the present worth, annual worth, and rate of return methods can be used when the effects of inflation are included, this lesson shows how the methods must be adjusted.
  • Comparisons with Risk
    This lesson extends the results of this section to the explicit consideration of risk. No deterministic analysis should be acceptable to the decision maker when there is significant uncertainty in the parameters of the analysis. The decision problem is made more difficult by risk, but the lesson shows how risk can be expressly included.

For most of this section, the problem is to select a solution from a set of mutually exclusive alternatives. Most of the materials in this section are in chapter 3 of the text.

Chapter 3: Engineering Economic Analysis
Present and Annual Worth

This lesson extends the present worth and annual worth evaluation methods to the comparison of alternative solutions. The extension is simple when the lives are equal. The best alternative is the one with the greatest NPW value. This assumes revenues are positive and costs are negative.

For the Net Present Worth Method compute the net present worth of the cash flows for the alternatives and choose the best. Be careful when the lives of the alternatives are not equal.

When the alternatives have different lives, a study period must be selected. A present worth comparison is meaningless unless the time represented by the present worth is the same for all alternatives. There are several ways to choose a study period, but we usually use the least-common-multiple of the lives. Replacements during the study period are like-for-like.

When the alternatives have different lives, they must be compared over a common study period. It is common to choose the least common multiple of the lives.

With the annual worth method, NAW values can be compared. Although it is not necessary to select a study period, the implicit assumption is the like-for-like replacements over the least common multiple of the lives. This makes the NAW method easier for comparisons with different lives. NPW and NAW comparisons always result in the same selection.

For the Net Annual Worth Method compute the equivalent uniform annual worth of the cash flows for the alternatives and choose the best.

Try the problems to test your skills. Don't look at the answers unless you solve the problem by yourself.

Comparison Problems
Comparison Exercise 1
Comparison Exercise 2

The homework for this section is in next lesson.

Rate of Return

This lesson considers the project selection problem and the comparison of alternatives problem. They are quite different in both statement and solution, but both use the IRR as a measure.

The project selection problem chooses a subset of the projects from a proposed set. For this problem compute the IRR for each alternative.

An organization has a set proposed projects from which some subset must be chosen. The IRR provides a measure for project selection.

When the MARR is given, all projects with IRR greater than the MARR are acceptable. The solution assumes there is no budget limit for capital expenditures.

With a given MARR and no budget restriction, select the projects that have IRR ≥ MARR

When there is a budget limit, use the selection rule below.

With restricted budget, rank the projects by IRR and in order of the ranking choose the largest subset that does not exceed the budget.

The rate of return method can also be used to select one solution from a set of mutually exclusive alternatives. Be sure to learn the incremental method. Generally, it is not necessary to calculate the IRR values for the individual alternatives. You must calculate the IRR for increments of investment. For examinations, you must follow the procedure completely in order to receive full credit.

To choose among mutually exclusive alternatives, you must use incremental analysis. The IRR of each increment of investment must satisfy the MARR requirement.

Try solving the problems with calculators in the toolbox before looking at the answers.

ROR Problems

Inflation complicates the evaluation of projects because dollars at different times have different buying power. This lesson introduces the language associated with inflation and provides tools for evaluating solutions.

With inflation there are two kinds of dollars, actual dollars and real dollars. There are also two values for the MARR. The market MARR evaluates cash flows expressed in actual dollars. The real MARR evaluates cash flows expressed in real dollars. Cost or revenue estimates are measured with today's prices. Escalation rates convert the estimates to actual dollars. The linked formulas calculate the economic measures for both real and actual dollars.

Economic Analysis Formulas

Since NPW is the equivalent at time zero, the value is the same whether the evaluation comes from real or actual dollars.

When the cash flow is in actual dollars use the market MARR to find the NPW. When the cash flow is in real dollars use the real MARR to find the NPW. The NPW values computed with the two methods are the same.

When component escalation rates are the same as the rate of general inflation, it is not necessary to change the estimates to actual dollars. Use the estimated cash flow and the real MARR to compute the NPW. This makes it easier to solve some problems.

When the escalation rates of all cash flow components are the same as general inflation, the estimated cash flow is the same as the real cash flow. Do not adjust it for inflation, but use the real MARR to find the NPW.

When inflation is present, comparison of alternatives may be more complex. When using the NPW method evaluate each alternative as indicated above and choose the best. For the NAW method, use real NAW values to compare the alternatives.

For the ROR method, use incremental analysis. When the cash flow is expressed in real dolloars, the IRR is a real rate of return. When the cash flow is expressed in actual dollars, the IRR is a market rate of return. When making decisions to accept or reject incremental investments, be sure to compare the IRR with the correct type of MARR. Real IRR values compare to the real MARR. Market IRR values compare to the market MARR.

Some the problems below include taxes. Neglect the parts that mention taxes because they are not considered in this course. Neglect the parts that mention taxes because they are not considered in this course. Skip 1.b, 3.c, and 6.c.

Inflation Problems

Skip 5 and 6.

Inflation Review

We extend the present and annual worth methods of comparison to include measures of risk. The primary measure of an alternative is it's mean NPW or NAW. When uncertainty is present, these measures are described by continuous probability distributions.

From the distributions we compute the risk measures that inform the decision maker about the risk associated with the NPW or NAW.

Risk Measures

To calculate the risk measures for an alternative, we must know its probability distribution or use simulation.

The NPW or NAW might be specified as a sum of independent random variables as below.

The moments are computed as the weighted sum of the component moments.

For a normal distribution, all the risk measures are computed from the given mean and standard deviation with the help of the cumulative and inverse of standard normal distribution. When the distribution is one of the named distributions, the risk measures can be computed with formulas or by using the functions of the Random Variables add-in.

When simulation is necessary, the risk measures can be computed from the simulated moments and by using the histogram of the NPW or NAW.

The risk measures provide additional information that the decision maker might use for the selection of the best alternative. With the normality assumption we may be able to eliminate solutions because of dominance. We sometimes use dominance even without the normality assumption.

The collection of undominated solutions is called the efficient frontier. Considering only the mean and standard deviation, the best solution is on the efficient frontier. If there is more than one solution on the frontier, we might construct a table as below showing the best and worst of each measure. The decision-maker must make the final choice.

Standard Deviation
 90% VaR


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

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