Selection

Consider this hypothetical situation. Brother is chronically short of cash and approaches sister with the following deal. Brother says "Hi sis. I need \$300 for a while. If you loan me \$300, I'll pay you back \$30 a month for the next year." How should sister respond?

She might wonder about brother's reliability for repayment, but for now we will neglect uncertainty in our analysis. A a later time, we will return to consider decision making with risk.

She might be concerned about mother's reaction. Mother doesn't like sister to make money from brother, and she is more generally concerned about the morality of charging interest for loans. Again we put politics, morality and other issues aside. This is not to say money is the most important issue, but sister wants to be well aware of the economics of the situation. She might adjust her decision later to keep family peace.

Although our example is personal, it is representative of decisions that are made every day in corporate and government settings. In this lesson we will see how the time value of money models considered earlier can be used to provide a single objective measure with which projects can be evaluated using either the minimum acceptable rate of return (MARR) or the minimum acceptable rate of borrowing (MARB), depending on the context. Once we have the measure, we will be able to answer whether to invest in a project or not. We will actually develop three equivalent measures: present worth, uniform worth and rate of return. Any one can be applied to a project and all give the same answer. We also investigate the payback measure that is widely used in practice. Later, we adjust the models to accommodate taxes and inflation. Both issues are important for the analysis of multiyear projects.

The methods of this lesson are used throughout the remainder of the course. In the lessons Comparisons with Net Worth and Comparisons with Rate of Return the methods are adapted to the comparison of alternative solutions to a given problem. In the lessons on Evaluation with Risk we show how the measures are used when uncertainty is explicitly recognized. The methods will also be used for in the Project Scheduling lessons.

This lesson is important to engineers. Projects proposed by engineers require the investment of capital funds, and the investment must be repaid with future revenues from sales or savings in operating costs. It is common corporate practice to require the engineer to justify the investment objectively and quantitatively. The methods of this lesson are often employed.

Click the QuickTime symbol to view an introductory presentation.

 Decision Making

Before going on, answer for yourself the question faced by sister. Should she accept brother's offer? Also answer the question: Is this a good deal for the brother? The answers are at the bottom of this page.

 Goals
 Given an investment or borrowing situation, use the NPW and NAW to determine if it is economically justified. Compute the payback period and use it for decision making. Use the Economics add-in for evaluating NPW and NAW.
 Text
 3.2 Compound Interest Formulas
 Procedure

This lesson considers the question of how organizations evaluate an investment opportunity with regard to its financial acceptability. In later lessons we use these methods to compare alternative solutions for a project and extend them to incorporate the effects of inflation, taxes and risk. The acceptability of a project is complicated because it involves a tradeoff between money spent or invested at one time in return for money received at a later time. The cash flow describing a project is multidimensional with positive and negative amounts at different points in time. To make decisions we will express the cash flow with a single equivalent measure.

 To decide whether an investment or loan is economically justified, express its cash flow with a single equivalent measure.

In this lesson we define three measures: present worth, uniform worth and payback period. The measures are used to accept or reject an investment or loan opportunity. The first two measures all result in the same conclusion, while the payback period is qualitatively different. We limit consideration to simple investments and simple loans in this lesson, while considering non-simple cases later. To determine whether a cash flow is simple, first find the net cash flow for each period. For projects that have several components for cash flow, the net cash flow at some time is the sum of all positive and negative cash flows that occur at that time. For a simple investment all negative cash flow amounts precede all positive cash flow amounts. For a simple loan all positive cash flow amounts precede all negative ones.

 For a simple investment all negative cash flow amounts precede all positive cash flow amounts. For a simple loan all positive cash flow amounts precede all negative ones.

The cash flows that we have seen to this point are all simple ones, but there are some important cases where non-simple cash flows arise

 Present Worth

The most common measure used to determine the acceptability of an investment is the net present worth (NPW). Click the QuickTime symbol to see the presentation. We use sister's decision problem to illustrate the NPW method for making investment decisions. Should sister make the investment in brother?

 Sister’s Dilemma

To maintain the same frame of reference, the main points of the lecture are generally stated in terms of investments. This is our emphasis for most of the course.

 The present worth method computes the NPW of the cash flow using the investor's MARR. If the NPW ≥ 0, accept the investment. Otherwise, reject the investment.

Brother has a decision similar to sister, but his is a borrowing situation. Should brother accept the loan from sister? Click the icon to see.

 Brother’s Plight

The method also helps in decisions related to loans.

 For a borrowing situation, the present worth method computes the NPW of the cash flow using the borrower's MARB. If the NPW ≥ 0, accept the loan. Otherwise, reject the loan.

The NPW method can be used in a variety of contexts. The next very short presentation summarizes the present worth method for making decisions.

 The NPW Method
 Annual Worth

For some people, the present worth measure is difficult to understand because its value represents the sum of present values for a number of periods in the future. The Annual Worth measure is more meaningful to business decision makers who are used to thinking in terms of annual financial statements. When the compounding period is smaller than one year, as the one-month interval in our example, the measure is the uniform worth per compounding period. For most organizational decisions the annual period is appropriate.

 The NAW Method

The net annual worth is a measure used to make decisions about investments when the interest used is the MARR. It can also be used to evaluate loans when the interest rate is the MARB.

 The annual worth method computes the equivalent net annual worth (NAW) of the cash flow over the life of the investment using the decision maker's MARR. If the NAW ≥ 0, accept the investment. Otherwise, reject the investment.

The annual worth may be computed directly from the cash flows, but when the NPW is known, the NAW is computed with a simple multiplication.

NAW = NPW(A/P, MARR, N)

Here N is the life of the investment expressed in compounding periods. The factor (A/P, MARR, N) is a positive number, so the sign of the NAW is the same as the sign of the NPW. Because the accept/reject decision is based on the sign of the measure, the decision is the same for the two methods.

 Computation

The Economics add-in provides convenient tools for defining and evaluating projects. Click the icon to see a QuickTime movie of the process of defining a project. The movie has no audio.

There are extensive instructions for the Economics add-in at the OM/IE site. Click the icon below to go to the instructions.

For the brother-sister loan proposal used to illustrate this case, we perform the analysis of the sister's cash flow on the Excel form below. Input data for the analysis are in the white cells and computed results are in the yellow and green cells. The yellow cells are computed with functions placed in the cell, so the student should not change these cells. The number in the green cell labeled IRR is computed by an algorithm provided by the add-in.

The project's Life, in D2, is expressed in years and the project's MARR, in F2, is expressed as a nominal value (per year). For sister this is 24% per year, or 2% a month. The entry in D4 shows that there are 12 compounding periods per year, because the loan payments are monthly. Other intervals on the form are expressed as periods, or months in this case. The cash flow data has two components. The loan amount of \$300 is entered as negative for the sister because it is an expenditure. It is a single payment. The payments of \$30 are receipts for the sister. These are paid as a uniform series over 12 months. The formulas in column I compute the present worth values of the components and the project. The project NPW is in cell I2. The NPW is positive, so this is an acceptable investment for the sister. We return to the other measures shown on the form later.

The brother's cash flow has a similar form, but the signs in the amount column are reversed. The other difference in the brother's form is the MARR. This is specified as a nominal rate of 36% per year, or 3% per month. The form uses the term MARR instead of MARB because it is primarily written for investments. Since I31 shows a positive value, the proposal is also acceptable to the brother.

The add-in also computes the equivalent uniform worth. We see the value of \$1.63 per month for sister in cell I3, and the value of \$0.14 per month for brother in cell I32. Both are positive, so the loan proposal is acceptable to both the brother and sister. As expected the decision is the same with both the NPW and NAW measures.

It is important to understand that the NPW measure (and the NAW measure) indicates the return of the investment relative to the MARR. If the NPW is zero, the investment returns exactly the MARR. When the NPW is positive, the investment earns more than the MARR, and when the NPW is negative it earns less than the MARR. When the NPW measure is zero or more, the investment is acceptable.

The add-in also computes a measure called the internal rate of return, or IRR. We discuss this measure as well as other features of the add-in in a later lesson.

The Computations section has a more complete example for evaluating projects with the Economics add-in.

 Payback

A final measure considered here is the payback period. It is often used in practice because it is simple and reflects several goals that are important to investors.

 The payback period is the number of years required for the returns attributed to an investment to equal to total amount of the investment.

When the compounding period is some other interval than years, the payback period is measured in compounding periods.

To compute the payback period it is necessary to know the cumulative cash flow over time. Our add-in computes the cumulative cash flow with the Show Cash Flow command. The figures below illustrate the cumulative values for the sister and brother. Note that the add-in has identified the sister and brother projects as a simple investment and a simple loan respectively. The sister's table shows that her capital investment is entirely recovered after 10 months. Similarly, brother's table shows that the loan principal is paid back after 10 months. The cash flows continue after 10 months because the payments include interest.

It happens that the investment of \$300 is exactly returned after 10 months. With discrete payments, there may be no whole number of periods that results in a cumulative cash flow of 0. Then a fractional payback period is computed using linear interpolation between the smallest positive value and the largest negative value. For example, if the loan payments were \$35 per month, the payback period would be 8.6 months.

 Sister's Cash Flow Brother's Cash Flow

The payback period is an important measure for capitalists because it indicates how soon the invested money is recovered from the project's receipts. Recovered money can be reinvested in some other opportunity. The payback measure gives no weight to cash flow values beyond the payback period. Since returns (and expenditures) are increasingly uncertain as they become more remote in time, requiring a short payback period is one way to control risk. Some companies accept opportunities with two or three year payback periods.

For simple investments the payback period is the time required for the initial investment to be recovered in cash amount from future receipts. An alternative concept is the discounted payback period. To compute this measure, the future cash flows are discounted with the MARR. The discounted payback period is the time required for the cumulative discounted cash flows to equal the initial investment. The discounted payback period is always longer than the payback period that does not discount the future cash flows.

Most textbooks on finance and engineering economics do not give much credence to payback as a measure because it neglects profitability. Clearly, examples can be constructed where there is zero profitability or very high profitability with the same payback period. For the current example, if the brother stops paying after 10 months, the sister makes no profit. If the brother makes the payments as planned but adds another \$300 to the final payment, sister will double her money. Both variations, as well as the original proposal, have the same payback period.

It is common practice in industry to have both payback and profitability hurdles. Only investments that pass both are acceptable.

We will not refer to the payback period during the remainder of the course. In deference to the importance the measure holds in practice, the student should know how to calculate it, and the add-in reports the payback on cash flow tables.

 The Answer to the Original Question

This discussion provided three answers to the original question. The present worth and uniform equivalent worth methods indicate that the loan arrangement is acceptable for both sister and brother. The payback method depends on the preferences of the decision makers, but both brother and sister have a ten-month payback.

 The sister with an MARR of 2% per month will accept and the brother with an MARB of 3% per month will also accept.

Since they are linearly related, the NPW and NAW measures always give the same answer. Of course, the answers depend on the MARR and MARB. Later we describe the rate of return method that, for simple investments, gives the same decisions as the NPW and NAW methods. The payback method neglects profitability, and there is really no reason why it would give the same answer as the others.

 Summary

 Selection Summary
 Problems

Click the icon for example problems with the answers provided.

 Economic Decisions

Engineering Finance
by Paul A. Jensen