Engineering Economics
Mubashir Rasheed
Multiple IRR
Values
 In the cash flow series presented thus far, the algebraic signs on
the net cash flows changed only once, usually from minus in year 0
to plus at some time during the series
 This is called a conventional ( or simple) cash flow series
 However, for some series the net cash flows switch between
positive and negative from one year to another, so there is more
than one sign change.
 Such a series is called nonconventional (nonsimple)
Multiple IRR
Values
 When there is more than one sign change in the net cash flows, it is
possible that there will be multiple i * values
 There are two tests to perform in sequence on the nonconventional
series to determine if there is one unique value or possibly multiple i
*values that are real numbers.
Multiple IRR
Values
 Test 1: (Descartes’) rule of signs states that the total number of realnumber roots is always less than or equal to the number of sign
changes in the series
 Test 2: Cumulative cash flow sign test, also known as Norstrom’s
criterion, states that only one sign change in a series of cumulative
cash flows which starts negatively indicates that there is one positive
root to the polynomial relation
 Zero values in the series are neglected when applying Norstrom’s
criterion
 Observe the sign of S0 and count the sign changes in the series S0 , S1
, . . . , S n . Only if S0 0 and signs change one time in the series is there
a single, real-number, positive i *.
Multiple IRR
Values
Example
Multiple IRR
Values
Example
Multiple IRR
Values
Example
Multiple IRR
Values
Example
Multiple IRR
Values
Example
EER: External
Rate of Return
 The reinvestment assumption of the IRR method may not be valid
in an engineering economy study. For instance, if a firm’s MARR is
20% per year and the IRR for a project is 42.4%, it may not be
possible for the firm to reinvest net cash proceeds from the
project at much more than 20% This situation, coupled with the
computational demands and possible multiple interest rates
associated with the IRR method, has given rise to other rate of
return methods that can remedy some of these weaknesses
 One such method is the ERR method
 If this external reinvestment rate, which is usually the firm’s
MARR, happens to equal the project’s IRR, then the ERR method
produces results identical to those of the IRR method
 In general, three steps are used in the calculating procedure.
 First, all net cash outflows are discounted to time zero (the
present) at ∈% per compounding period.
 Second, all net cash inflows are compounded to period N at ∈%.
 Third, the ERR, which is the interest rate that establishes
equivalence between the two quantities, is determined.
EER: External
Rate of Return
 The absolute value of the present equivalent worth of the net cash
outflows at ∈% (first step) is used in this last step. In equation
form, the ERR is the i′% at which
 In general, three steps are used in the calculating procedure.
 First, all net cash outflows are discounted to time zero (the
present) at ∈% per compounding period.
 Second, all net cash inflows are compounded to period N at ∈%.
 Third, the ERR, which is the interest rate that establishes
equivalence between the two quantities, is determined.
EER: External
Rate of Return
 The absolute value of the present equivalent worth of the net cash
outflows at ∈% (first step) is used in this last step. In equation
form, the ERR is the i′% at which
EER: External
Rate of Return
Example
EER: External
Rate of Return
Example
 When two or more mutually exclusive alternatives are evaluated,
engineering economy can identify the one alternative that is the
best economically
 Incremental Analysis technique is used to selected the best
alternative among the presented
IRR of Multiple
Alternatives
 When independent projects are evaluated, no incremental
analysis is necessary between projects. Each project is evaluated
separately from others, and more than one can be selected.
IRR of Multiple
Alternatives
 To conduct an incremental ROR analysis, it is necessary to
calculate the incremental cash flow series over the lives of the
alternatives.
IRR of Multiple
Alternatives
Incremental
analysis
 The incremental ROR method requires that the equal-service
requirement be met. Therefore, the LCM (least common
multiple) of lives for each pairwise comparison must be used. All
the assumptions of equal service present for PW analysis are
necessary for the incremental IRR method.
 When a study period is established, only this number of years is
used for the evaluation. All incremental cash flows outside the
period are neglected
 Only for the purpose of simplification, use the convention that
between two alternatives, the one with the larger initial
investment will be regarded as alternative B .
 Incremental cash flow = cash flow B - cash flow A
IRR of Multiple
Alternatives
Incremental
analysis
IRR of Multiple
Alternatives
Incremental
analysis
IRR of Multiple
Alternatives
Incremental
analysis
IRR of Multiple
Alternatives
Incremental
analysis
 The incremental cash flows in year 0 of Tables 8–2 and 8–3 reflect
the extra investment or cost required if the alternative with the
larger first cost is selected
IRR of Multiple
Alternatives
Incremental
analysis
 If the incremental cash flows of the larger investment don’t justify
it, we must select the cheaper one. In Example the new drill press
requires an extra investment of $6000 (Table 8–2). If the new
machine is purchased, there will be a “savings” of $1200 per year
for 25 years, plus an extra $300 in year 25. The decision to buy the
used or new machine can be made on the basis of the profitability
of investing the extra $6000 in the new machine. If the equivalent
worth of the savings is greater than the equivalent worth of the
extra investment at the MARR, the extra investment should be
made (i.e., the larger first-cost proposal should be accepted). On
the other hand, if the extra investment is not justified by the
savings, select the lower-investment proposal
 If the rate of return available through the incremental cash flow
equals or exceeds the MARR, the alternative associated with the
extra investment should be selected.
IRR of Multiple
Alternatives
Incremental
analysis
 For multiple revenue alternatives, calculate the internal rate of
return i * for each alternative, and eliminate all alternatives that
have an i * MARR. Compare the remaining alternatives
incrementally
 When independent projects are evaluated, there is no
comparison on the extra investment. The ROR value is used to
accept all projects with i * MARR, assuming there is no budget
limitation.
IRR of Multiple
Alternatives
Incremental
analysis
Example
IRR of Multiple
Alternatives
Incremental
analysis
Example