Chapter 7A Difficulties Solving for the IRR Click here for Streaming Audio To Accompany Presentation (optional) Click here for Streaming Audio To Accompany Presentation (optional) EGR 403 Capital Allocation Theory Dr. Phillip R. Rosenkrantz Industrial & Manufacturing Engineering Department Cal Poly Pomona

EGR Cal Poly Pomona - SA102 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. 10, 11 - 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 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)

EGR Cal Poly Pomona - SA109 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%