Appendix 7A Difficulties Solving for the IRR EGR 403 Asset Allocation in Technical Decision Making Dr. Phillip R. Rosenkrantz Industrial & Manufacturing Engineering Department Cal Poly Pomona Based on Newnan © 2009 by Oxford University Press, Inc.

EGR Cal Poly Pomona - SA10 Revised2 EGR 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 Replacement Analysis

EGR Cal Poly Pomona - SA103 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

EGR Cal Poly Pomona - SA104 Example 7A-1 This series of cash flows produces two solutions for IRR: 10.2% and 47.3%.

EGR Cal Poly Pomona - SA105 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 Number of positive values of X Number of sign changes, m

EGR Cal Poly Pomona - SA106 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 Cal Poly Pomona - SA107 Zero Sign Changes Receiving a gift. Giving your friend a loan and not being paid back. In either case no “i” can be computed.

EGR Cal Poly Pomona - SA108 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, R i, or net expense, E i 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.