Engineering Finance

Project Evaluation
Rate of Return

Non-Simple Cash Flow

A non-simple cash flow is shown to the left. It is characterized by more than one change in sign, or more than one reversal, in the cash flow amounts. Exact amounts are shown in the table below. There are five sign changes as every other amount has a different sign. As an alternative to the term non-simple we also call this a mixed cash flow, because at certain times the cash flow is an investment and other times it is a loan.

A mixed cash flow presents problems in computing the IRR in that the NPW(i) function may have multiple roots. There may be as many roots as the number of cash flow reversals. A simple investment or loan has one reversal and at most one root. The IRR is the value of that root if it is positive. The cash flow at the right has three positive roots. This can be seen in the NPW function plotted below.

The three figures below show the cash flow table and the IRR analyses that identify the three roots as 7.13%, 44.04%, and 92.47%. The IRR is computed with the Excel function: IRR(values, guess). The numbers in the cash flow range provide the "values" argument, and the "guess" argument is in the cell immediately above the IRR. For instance, the first case uses a guess of 10%, the second a guess of 30% and the third a guess of 90%. We discuss the RIC measure later on this page.

The existence of three different values complicates the accept-reject decision. The IRR does not depend on the MARR, but the decision depends on the MARR. When the MARR is 10% the first value rejects the investment while the other two accept it. What is the correct answer?

Use the Present Worth?

Some authors suggest that the IRR is unreliable in the event of mixed cash flows, so the present worth or annual worth values should be used. The figure above shows that when the MARR is 10% the NPW is negative, so the project should be rejected. This is logically unsatisfying because we note that if the MARR were 50% the project would be accepted. How can a project be unacceptable at one MARR but be acceptable at some much greater MARR? Although the present worth and annual worth methods give a single answer, the answer in this sense is not logical.

Use the RIC

The return on invested capital, RIC, is used by the add-in to find a single interest measure that depends only on the cash flow and what we call the external investment rate. The external investment rate is the rate that the organization can earn on money external to the project. A reasonable value to use for this is the MARR. One definition of the MARR is "the rate that you could earn if the money were not invested in the project".

The difficulty with the example cash flow is that the project is mixed in that at some time in its life it uses cash from the organization (an investment), while at others it is loaning cash to the organization (a loan). All three measures, NPW, NAW and IRR, compound the current balance with the same interest rate whether the balance is positive or negative.

Let's say you have an arrangement with a bank that allows you to borrow money and deposit money in the same account. When the account is negative, that is the bank is investing in you, you must pay interest to the bank. When the account is positive, you are loaning money to the bank and they will pay interest to your account. If your bank is like mine, they will not charge the same interest rate in the two cases. The bank will charge a higher rate when you owe them money, and pay a lower interest rate when it owes you money.

This is the same as the RIC computation. The computation produces a cumulative balance as the right-most column in the figure above. Here we call the balance the Net Investment. When the balance is negative, the project is an investment and the balance grows (becomes more negative) with interest rate i. When the balance is positive, the project is loaning money to the sponsor, so the balance grows (becomes more positive) with interest rate equal to the MARR. The RIC is the value of i that makes the cumulative value at the last period equal to zero. This brief discussion is only an introduction to a complicated concept. The important point here is that RIC gives one value when the IRR may give more than one.

The value of the RIC is unique and it leads to the same conclusion as the NPW method. The add-in assumes that the external rate of return is equal to the MARR. For the example when the MARR is 10%, the RIC is 9.65%. Both the NPW and the RIC reject the investment.

A number of other measures have been suggested that yield a single rate for evaluating a project. One is the modified internal rate of return, MIRR. This is discussed at several sites on the web, and is implemented in the MIRR Excel function.


When evaluating simple investments, the NPW, NAW and IRR measures give the same accept/reject decision. Each of these equivalent measures are useful in different contexts. The RIC measure is an alternative to the IRR for mixed investments. The payback period is used widely in practice, but bears little relation to the others since it neglects profitability.

We should note that there are other measures that one could use to decide whether a particular project is financially acceptable. Goldratt in the popular book, The Goal, suggests that the profitability measures described here are not sufficient. He argues that the benefits and costs for a process improvement project cannot be isolated correctly for the analysis methods we have described. Rather, he provides three different financial measures: throughput, cash flow and inventory. The interested reader is advised to study this alternative view.

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

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