Chapter 7A
Difficulties Solving for
the IRR
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EGR 403 Capital Allocation Theory
Dr. Phillip R. Rosenkrantz
Industrial & Manufacturing Engineering Department
Cal Poly Pomona
EGR 403 - The Big Picture
• Framework: Accounting & Breakeven Analysis
• “Time-value of money” concepts - Ch. 3, 4
• Analysis methods
– Ch. 5 - Present Worth
– Ch. 6 - Annual Worth
– Ch. 7, 7a, 8 - Rate of Return (incremental
analysis)
– Ch. 9 - Benefit Cost Ratio & other techniques
• Refining the analysis
– Ch. 10, 11 - Depreciation & Taxes
– Ch. 12 - Replacement Analysis
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Multiple IRR
Occurs when a cash flow produces more than one point
at which NPW = 0. This happens when there is more
than one sign change in the cash flow series
Example 7A-1
Cash Flow
Year
0
1
2
3
4
5
Cash Flow
19
10
-50
-50
20
60
80
60
40
20
0
-20
-40
-60
1
2
EGR 403 - Cal Poly Pomona - SA10
3
4
5
6
3
Example 7A-1
This series of cash flows produces two solutions
for IRR: 10.2% and 47.3%.
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Cash Flow Rule of Signs
• This happens when we convert the IRR
equation to a polynomial.
• Then, by Descartes’ rule
Number of sign
changes, m
0
1
Number of positive
values of X
0
1
2
3
2 or 0
3 or 1
4
4, 2 or 0
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Cash Flow Rule of Signs Expands
on This Notion
• There may be as many positive values of “i” as
there are sign changes in the cash flow.
• Sign changes are counted when:
• + To -.
• - To +.
• A zero cash flow is ignored.
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Zero Sign Changes
• Receiving a gift.
• Giving your friend a loan and not being
paid back.
In either case no “i” can be computed.
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Solving for ROR
• We use an “external rate of return” to adjust
cash flows so that we have only one sign
change.
• External interest rate is almost like a money
market rate and is different than the MARR.
• Move the least amount of positive cash flow
forward that you can to eliminate all but one
sign change. (Note: Cannot move negatives
cash flows forward)
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Example 7A-2: Solving for a more
realistic IRR
• We have two sign changes.
• The easiest way to reduce that to one is by moving the cash flow in
years 0 and 1 to year 2.
• Use the “external interest rate” to move the two cash flows ahead
• 8.4% is a more realistic IRR for this project than 10.2% or 47.3%
Year
0
1
2
3
4
5
Cash Flow
External Interest Rate = 6%
Revised
19
Move two years: 19 (F/P, 6%, 2) = 21.40
0
10
Move one year: 10 (F/P, 6%, 1) = 10.60
0
-50
21.4 + 10.6 = 32
-18
-50
-50
20
20
60
60
IRR = 8.4%
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