Review

This is a review for the second exam. The purpose is to summarize the contents of lessons 9 - 16, which contain the material that we have studied since the first exam. You should also review the event and homework keys on the class Canvas site.

A final suggestion is to visit the Presentations section that can be found under the Resources tab. There, the QuickTime movies are listed in order for easy viewing. For the example problem in this lesson, use the calculators in the Toolbox to find the solutions.

 Toolbox
 Goals

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.

 9. Time Value of Money The lesson describes how compounding, time and interest rate are used to express an amount of money at one point in time as an equivalent amount at another. These ideas are central to the procedures used to economically evaluate and compare project solutions. 10. Equivalence Factors The subjects of Engineering Economics and Business Finance traditionally use equivalence factors to express a complicated cash flow as a single equivalent number. This lesson introduces these factors and shows how they can be used to answer a variety of interesting business and personal finance questions. 11. Net Worth For investment analysis, common economic measures are the net present worth and net annual worth of a cash flow. This lesson shows how these measures are computed using the equivalence factors. 12. Selection To accept or not accept is the question posed for project selection. This lesson shows that the answer depends on the investor's minimum acceptable rate of return (MARR). The lesson provides a simple rule to make the decision. 13. Comparisons with Net 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. 14. Rate of Return The rate of return discussed in this lesson gives the same answers as the net worth methods, but it is more familiar to most decision makers. The lesson shows how to calculate the internal rate of return (IRR) for a cash flow and how it is used to make selection decisions. 15. Comparisons with Rate of Return This lesson explores the rate of return method for comparing alternatives. We learn the important principle that the rate of return method must be applied to increments of investment rather than directly to the individual alternatives. 16. Inflation Inflation of costs and revenues is inevitable in modern economies, measurabley complicating 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.
 Exam Format

This midterm exam will be administered during the regular class meetings. The exam will have both closed and open book parts. The closed-book part will be distributed first. When you turn in the closed-book part, you will receive the open-book part. No strict time limit is enforced for the completion of the closed book-part, but leave at least one hour for the open-book part. The open-book part must be submitted by the end of the class meeting. You will have 1.75 hours to complete both parts.

 Text

Chapter 3 in the textbook covers most of the topics for this exam.

 3.1-3.5: Engineering Economic Analysis

Click the link on the title above to open the lesson. Use the back button to return to this review.

This is an introductory lesson to show how money grows with interest. Review the flash movie to see how interest affects growth and the difference between simple and compound interest.

 Growth with Interest

The lesson also introduces discounting as the inverse of compounding. Be sure you know what both mean. Be able to say why invested money grows even if there were no inflation.

Equivalence factors move money around. The purpose for doing that is to replace a complex cash flow with a single equivalent. With a single measure, solutions can be judged for economic acceptability. In the next section we use them to compare alternative solutions. Factors are also useful for computing quantities related to loans and investments.

Review the formulas that define the equivalence factors. The factors are derived for end-of-period payments. You don't have to know these formulas but be aware of the timing of the cash flows assumed for each factor.

 Factor Formulas

When the interest rate is zero, the single value factors simply compute the sum of the amounts in the cash flow. The distribution factors such as A/P or A/F divide a single value by the number of payments. When the number of periods goes to infinity, some of the factors have limiting values. Review the limit values of the factors.

 Limit Values for Factors

The normal compounding period is one year, but when there are several compounding periods in a year, it is often necessary to compute the effective rate.

 Effective Interest Formulas

The payments and payoff values for installment loans can be computed with simple applications of equivalence factors. For loans, the interest rate, number of periods, initial principal and the amount of payment are well defined. Given any three of these quantities, the other can be determined. Review the use of the formulas for loans.

 The Economics of Loans

Application of a single factor can answer a variety of interesting questions. Review the Simple Time Value problems given in Lesson 10.

 Simple Time Value

In Lesson 11 we use the equivalence factors to find the net present worth and net annual worth of given cash flows. For simple problems this course adopts the goal of using the fewest time value of money factors as possible to express the present or annual worth. This involves moving money around. Review the Flash movie to see some examples.

 Moving Money Around

The cash flow exercises show more examples including some with periodic payments. Answer the questions and use the scorecard and the answer key to see how you did.

 Cash Flow Exercises
 Scorecard
 Solutions for Cash Flow Exercises

These problems show some simple practical situations that are solved by finding the present or annual worth.

 Time Value of Money

Lesson 12 applies the equivalence factors to decision making with the goal of deciding whether to accept or reject the cash flow for a particular solution. Of primary importance are are the minimum acceptable rate of return (MARR) for an investment situation and the maximum acceptable rate of borrowing (MARB) for a borrowing situation. We primarily focus on investments because these are most often required for engineering projects.

Some simple problems are illustrated below.

 Economic Decisions

This lesson extends the present worth and annual worth evaluation methods to the comparison of alternative solutions. When 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 NPW 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 NAW 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 until you have tried to solve the problem by yourself.

 Comparison Problems
 Comparison Exercise 1
 Comparison Exercise 2

The following link opens short-answer questions regarding the use of the present and annual worth measures.

 Present/Annual Worth Review

The Economics add-in is easy to use and is one of the tools available for open-book exams. For a review of the add-in basics without audio, click on the QuickTime icon.

For lesson 14 we provide a new measure called the rate of return of the investment or the internal rate of return (IRR). This value is computed by solving for the interest rate that makes the NPW or NAW equal to zero.

Click the QuickTime symbol to see the introductory presentation for rate of return comparisons.

 Rate of Return Method

 The IRR is the interest rate for which the NPW (or NAW) calculated with the IRR is exactly 0. If the IRR ≥ MARR, accept the investment; otherwise, reject it.

It is good to be able to recognize the simple cases for the IRR calculations. These will often save time over the alternative of trial and error.

 Simple Cases for IRR

Students should be able to demonstrate the trial and error approach for finding the IRR. This approach is not too difficult for a simple investment since the NPW is a decreasing function of i. When searching for the solution if one computes a positive NPW for some i, the IRR must be greater than i. If one computes a negative NPW, the IRR must be smaller than i.

The IRR search process seeks two different interest rates, one with a positive NPW and one with a negative NPW. The IRR estimate is found with linear interpolation by computing the approximate zero-crossing point. When demonstrating trial and error the student cannot use an automatic numerical solver such as the one built into Excel or implemented into the Economics add-in.

For simple investments (with positive profit) there is a single solution for IRR. For mixed (or non-simple) investments there may be more than one solution. A second measure called return on invested capital (RIC) provides a single solution for all problems, so it can be used for decision making.

The problems illustrate some ROR problems.

 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

The following link opens short-answer questions regarding the use of the rate of return method for evaluation and comparison.

 ROR Review

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 expressed in 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

Because 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 dollars, 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.

 Inflation Problems

The link opens short-answer questions regarding the analysis of the effects of inflation.

 Inflation Review

For a review of the Economics add-in for problems with inflation, click on the QuickTime icon. This movie has no audio.

 Economics add-in with Inflation
 Summary

 Project Evaluation Summary

Engineering Finance
by Paul A. Jensen