Appendix 7ADifficulties Solving for

the IRR

EGR 403 Asset Allocation in Technical Decision Making

Dr. Phillip R. RosenkrantzIndustrial & Manufacturing Engineering Department

Cal Poly Pomona

Based on Newnan © 2009 by Oxford University Press, Inc.

EGR 403 - Cal Poly Pomona - SA10 Revised 2

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. 11, 12 - Depreciation & Taxes– Ch. 13 - Replacement Analysis

EGR 403 - Cal Poly Pomona - SA10 3

Multiple IRROccurs 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-1Cash Flow

-60

-40

-20

0

20

40

60

80

1 2 3 4 5 6

Year Cash Flow0 191 102 -503 -504 205 60

EGR 403 - Cal Poly Pomona - SA10 4

Example 7A-1This series of cash flows produces two solutions for IRR: 10.2% and 47.3%.

EGR 403 - Cal Poly Pomona - SA10 5

Cash Flow Rule of Signs• This happens when we convert the IRR

equation to a polynomial. • Then, by Descartes’ rule

4, 2 or 04

3 or 13

2 or 02

11

00

Number of positive values of X

Number of sign changes, m

EGR 403 - Cal Poly Pomona - SA10 6

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.

EGR 403 - Cal Poly Pomona - SA10 7

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 (a.k.a. IRR)We will use the Modified Internal Rate of Return (MIRR) method

to adjust cash flows so that we have only one sign change.

• Combine cash flows in each period to get a single net Receipt, Ri, or net expense, Ei

• Find the present worth of the expenses with the external financing rate (rate for borrowing money)

• Find the future worth of the receipts with the external investing rate (rate for invested money).

• Find the MIRR which makes the present and future worths equivalent

• If the MIRR is less than the external investing rate, it is not desirable

Engineering Economic Analysis

Newnan Copyright © 2009 by Oxford

University Press, Inc.

Figure revisited ex07A-02.

FIGURE 7A-3 PW versus i for oil well.

FIGURE 7A-5 MIRR for the oil well.

FIGURE 7A-6 MIRR for oil well.