Internal Rate of Return Internal Rate of Return (IRR): The interest rate i* at which NPW = 0 Note: This is the same as finding the roots of a polynomial equation. If there is more than one sign change in the net annual cash flows over the life of the project, then there is more than one internal rate of return (root)! We may find the IRR by either the manual method we used for the bond yield, or we 1 may use the computer to find the roots by IRR Example Project A costs $10,000 and will last for 10 years. Annual, end of year revenues will be $3000, and expenses will be $1000. There is no salvage value. Project B costs $20,000 and will also

last for 10 years. Annual revenues will be $4000 with annual expenses of $1500. Salvage value is $14,500. What is each projects IRR? 2 Table Example i* (P/A, i*, 10) (P/F, i*, 10) 6% 7.3601 .5584 8%

6.7101 .4632 10% 6.1446 .3855 12% 5.6502 .3220 18% 4.4941 .1911

Numerical Method Example: IENG 302 IRR.xls 3 25000 Project A Net Present Worth ($) 20000 Project B 15000 10000 5000 0 0 5 10

15 20 -5000 -10000 Interest Rate (% ) 4 Decisions with IRR When applied to project selection among independent projects i.e. there are enough funds such that any or all of the projects may be selected then investing based on IRR is easy: Select all projects with IRR > MARR!

5 Decisions with IRR When applied to project selection among mutually exclusive projects i.e. there is not enough money to do them all IRR can produce the same result as NPW and EAW. However, incremental analysis MUST be used! REASON: IRR is a relative measure of project merit 6 Why Must Incremental Analysis be Used for Competing Projects? Assume that an MARR of 16% per year is required, and $85 000 is available

to invest: Project A requires $50 000 upfront to obtain an IRR of 35% per year. Project B requires an $85 000 first cost and returns an IRR of 29% per year. What could we do with the uninvested money from Project A? ($35 7 000) Why Must Incremental Analysis be Used for Competing Projects? It would be reasonable to invest the remaining $35 000 at the MARR: Overall IRRA = 50 000(.35) + 35 000(.16) 85 000 = 27.2% per year Project B returns an IRR of 29% per year on ALL the money available to invest. 8

Incremental Analysis A technique or approach that can be used with NPW, EAW, and later with IRR and Cost/Benefit to determine if an incremental expenditure should be made. Note: If using NPW or IRR, lifetimes must be equal so use Least Common Multiple or Study Period! 9 Incremental Analysis Incremental Analysis can used by putting the options in the order of increasing initial investment. Check the feasibility of the first investment. If it is feasible, then you consider the increment to the next level of

investment. If it is not feasible, then you check the second investment alternative for feasibility, and so on until all 10 options are considered. Incremental Analysis Incremental Analysis can also be used when youre already incurring an expense (e.g., DSL internet service) and you are trying to determine if it is a good decision to spend additional funds (e.g., satellite or cable?). 11 Steps of the Process 1. Order alternatives from lowest to

highest initial investment. 2. Let Alternative A0 (do nothing) be considered the current best. 3. Consider next Alternative ( j = j+ 1) 4. Determine cash flows for current best and Alternative j. 5. Determine incremental cash flows between current best and Alternative j. 6. Calculate PW, AW, FW, IRR (or Benefit/Cost) 12 of only the incremental cash flows. Steps of the Process 7. If incremental investment yields NPW, EAW, or a NFW > 0*, then the new current best becomes Alternative j. * (B/C ratio > 1, or IRR > MARR) 8. If there are remaining

alternatives, go to Step 3. 9. If all alternatives have been considered, select the current best alternative as the preferred alternative. 13 i = 18% Net Cash Flows for Alternatives A 0 - A3 End of Year, t 0 A0 A1 A2 A3

$0 $0 -$50,000 -$75,000 1 0 4,500 20,000 20,000 2 0

4,500 20,000 25,000 3 0 4,500 20,000 30,000 4 0 4,500

20,000 35,000 5 0 4,500 20,000 40,000 14 IRR Incremental Rank projects from lowest to highest Analysis initial cost

Eliminate any projects with IRR < MARR Starting from the least expensive project to the next most expensive, justify each incremental investment Accept LC IRR HC-LC < MARR Project & Reject HC Project IRR HC-LC = MARR Indifferent IRR HC-LC > MARR Project Reject LC 15