CE 203 Rate of Return Analysis (EEA Chapter 7)

1. CE 203Rate of Return Analysis(EEA Chapter 7)

2. Rate of Return Analysis • “Equivalent” cash flows: same value at some given time for a given interest rate • Internal rate of return (definitions): • interest rate such that, for given payment schedule, loan is paid off with final payment • interest rate such that, for given payment schedule, unrecovered investment = 0 at final payment • interest rate such that benefits = costs

3. Rate of Return Analysis, RoR • P = F (P/F, i, n) or P = A(P/A, i, n) EEA 5 • A = P (A/F, i, n) EEA 6 EEA 7: i? for benefits = costs

4. Rate of Return Analysis • Internal RoR, i*, solve for i in : • NPW = PWB – PWC = 0 • EUAW = EUAB – EUAC = 0 • To solve for i* : • Iterative solution (get close, interpolate) • Use “solver” • Plot NPW or EUAW, “read” i* at NPW = 0 • Spreadsheet (Excel or ???) • RATE (N, A, P, F, Type, guess) • IRR (value, guess)

5. In-class example 7-1 If you invest $10,000 now and are paid $5,200 at the end of each of the next two years, what is the internal rate of return? Use iteration, then interpolation to find i NPW = $ 5,200(1+i)-1 + $ 5,200(1+i)-2 - $10k = 0 Try 2% = $ 5,098 + $ 4,998 - $ 10,000 = $ 96 Try 3% = $ 5,045 + $ 4,902 - $ 10,000 = -$ 53 Interest rate, from linear interpolation 2% + 96/(96+53)(3-2) = 2.64% OR: Use SOLVER

6. In-class example 7-1 Use plotting:

7. Rate of Return Analysis • Chapter 7: compare two alternatives • Chapter 8: compare three+ alternatives • Advantages of RoR analysis: • More widely understood • Single value of “merit” • Most widely used (but maybe not in CE?)

8. What is easier to understand? • NPW = $5000 • EUAW = $800 • RoR = 8%

9. Investment vs. Borrowing Situation • Investment: subsequent inflow > initial amount • Borrowing: subsequent outflow > initial amount • Usually (but not always) investigate initial cash flow • Investment if negative • Borrowing if positive

10. Investment vs. Borrowing Example Investment Borrowing Sum = $1,000 Investment Sum = - $1,000 Borrowing

11. Minimum Attractive Rate of Return • Minimum Attractive Rate of Return (MARR) Rate of return (RoR) below which we will not invest (because we can invest elsewhere at MARR or simply decide not to invest if RoR is < MARR) • MARR is the highest of • Interest rate for borrowing money • Average interest rate for the cost of capital (loans, bonds, stock, etc.)

12. Rate of Return Analysis • RoR criterion:If internal rate of return (i*) P> MARR, the investment is considered acceptable (but not necessarily the best) • RoR analysis for two alternatives • Determine the cash flow for the difference between alternatives (highest total cash flow alternative minus lower total cash flow alternative) • Determine the incremental rate of return (DIRR) on the difference between the alternatives and compare to MARR • If DIRR> MARR, choose higher-cost alternative • If DIRR< MARR, choose lower-cost alternative

13. In-class Example 7-2 Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%. (RoR = 14.5%) (RoR = 11.9%) Which is the better alternative? To answer, consider both RoR and MARR

14. In-class Example 7-2 Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%. Both total cash flows are positive, so both are “investments”; Alternative 2 has larger total, so use Alt. 2 – Alt. 1 for DIRR

15. In-class Example 7-2 Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%. Note: total or net cash flow for difference is positive

16. In-class Example 7-2 Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%. Need i such that NPW = 0 = - 4000 (1 + i) -1 + 4300 (1 + i) -2For this simple case, i = 300/4000 = .075 = 7.5% = DIRR

17. In-class Example 7-2 Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%. Since DIRR = 7.5% > MARR = 6%, choose alternative #2

18. In-class Example 7-2 Or, to look at it another way: Alternative #1 Alternative #2

19. In-class Example 7-2 with MARR = 9% Suppose MARR = 9% for payback alternatives for an initial investment of $5000: Since DRoR = 7.5% < MARR = 9%, choose alternative #1

20. In-class Example 7-2 with MARR = 9% Alternative #1 Alternative #2

21. In-class Example 7-2 but we are the borrower First, if MARR = 6%, we would choose neither alternative and go to bank to get $$$. If forced to choose, selection criterion is reversed: we would choose Alternative #1.

22. In-class Example 7-3 What is the internal RoR (i*) for the cash flow shown in the table below? 0 = 1020 – 2000(P/F, i, 1) + 500(P/F, i, 2)+ 500(P/F, i, 3). Solve for i.

23. In-class Example 7-3 0 = 1020 – 2000(P/F, i, 1) + 500(P/F, i, 2)+ 500(P/F, i, 3). Graphing solution (using EXCEL): NPW = 0 at i values of 5.24% and 27.4% Two answers!

24. Multiple values for ROR possible! …there may be as many positive values for i* as there are sign changes in cash flow table(in example, +1020 to -2000 to +500) …try the modified internal rate of return (p. 238 of the textbook)