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
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.
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.
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
decide whether an investment or loan is economically
justified, express its cash flow with a single equivalent
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.
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
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
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?
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.
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.
The method also helps in decisions related to loans.
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.
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
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
For most organizational decisions the annual period is appropriate.
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.
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
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.
Add-in: Defining a Project
There are extensive instructions for the Economics add-in at the OM/IE site. Click the icon below to go to the
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.
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
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
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.
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
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.
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
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.
Click the icon for example problems with the
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