Engineering Economic Analysis

232 RATE OF RETURN ANALYSIS

Thus, Descartes' rule for polynomials gives the following:

Number of Sign

Changes,m

o

1

2

3

4

Number of Positive

Values ofx

o

1

20rO

3 or 1

4, 2, or 0

N~ber of Positive

Values ofi

o

10rO

2, 1, or 0

3, 2, 1, or 0

4, 3, 2, 1, or 0

Ifxis greater than I, the corresponding value ofiis negative. If there is only one root

and it is negative, then it is a valid IRR. One example of a valid negative IRR comes from

a bad outcome, such as the IRR for a failed research and development project. Another

example would be a privately sponsored student loan set up so that if a student volunteers

for the Peace Corps, only half of the principal need be repaid..

If a project has a negative and a positive root fori,then for most projects only the

positive root ofiis used..

Projects with Multiple Sign Changes

Example 7A-I is representativeof projects for which there are initial payments (e.g., when

the order for the ship, airplane, or building is signed), then the bulk of the costs occur, and

then there are more payments on completion..

Projects of other types often have two or more sign changes in their cash flows.Projects

with a salvage cost typically have two sign changes. This salvage cost can be large for

environmentalrestoration at termination. Examples include pipelines, open.pit mines, and

nuclear power plants. The following cash flow diagram is representative.

$50K

t t t t t t

0-1-2-3-4-5-6-7

j ~tK

$180K

Many enhancement projects for existing mines and deposits have a pattern of two

sign changes. Example 7A-2 describes an oil well in an existing field. The initial invest-

ment recovers more of the resource and speeds recovery of resources that would have

been recovered eventually.The resources shifted for earlier recovery can lead to :twosign

changes.

In Example 7A-3, we consider staged construction, where three sign changes are

common.