Chapter 8 Choosing the Best Alternative 1 Chapter

Chapter 8 Choosing the Best Alternative 1

Chapter Outline Incremental Analysis Graphical Technique in Solving problems with Mutually Exclusive Alternatives Using Spreadsheets in Incremental Analysis 2

Learning Objectives Define Incremental Analysis Apply Graphical Technique in Solving Problems with Mutually Exclusive Alternatives Use Spreadsheets in Incremental Analysis 3

Internal Rate of Return (IRR) By definition in Chapter 7: Given a cash flow stream, IRR is the interest rate i at which the benefits are equivalent to the costs. This can be expressed mathematically in five different ways. NPW=0 PW of benefits - PW of costs =0 PW of benefits = PW of costs PW of benefits / PW of costs=1 EUAB-EUAC=0 4

Example Cash flows for an investment are shown in the following figure. What is the IRR to obtain these cash flows? YEAR CASH FLOW 0 ($500) 1 $100 2 $150 3 $200 4 $250 5

QUESTION CONTINUES -8. 85 YEAR CASH FLOW 0 ($500) 1 $100 2 $150 3 $200 4 $250 6

QUESTION CONTINUES -8. 85 YEAR CASH FLOW 0 ($500) 1 $100 2 $150 3 $200 4 $250 7

INTERPOLATION 10 5 -X 5% 30. 95 X% 0 15% -8. 85 30. 95 39. 80 8

INTERPOLATION 10 5 -X 5% 30. 95 X% 0 15% -8. 85 30. 95 39. 80 9

EXCEL solution IRR = irr(a 1: a 5) = 12. 83% 10

Mutually Exclusive Alternatives Only one alternative may be implemented All alternatives serve the same purpose Objective of incremental analysis is to select the best of these mutually exclusive alternatives 11

Incremental Analysis When there are two alternatives, rate of return analysis is performed by computing the incremental rate of return, ΔIRR, on the difference between the two alternatives, as discussed in Chapter 7. 12

Incremental Analysis The cash flow for the difference between alternatives is calculated by taking the higher initial-cost alternative minus the lower initial-cost alternative. The below decision path is made for incremental rate of return (ΔIRR) on difference between alternatives: Two -Alternative Situations Decision ΔIRR≥MARR Choose the higher-cost alternative ΔIRR<MARR Choose the lower-cost alternative 13

Example The cash flows for four different alternatives are given in table below. If MARR 10%, which is the best alternative? Using the incremental analysis, we need to repeat 3 times, by comparing 2 alternatives at a time. 14

MARR = 10% EUAB =EUAC (Increment) ΔIRR≥MARR ΔIRR<MARR Choose the higher-cost alternative 15 Choose the lower-cost alternative

Incremental Analysis Could be applied to rate of return (IRR), present worth (PW), equivalent uniform annual cost (EUAC), or equivalent uniform annual worth (EUAW) approaches. [Higher-cost alternative] = [Lower-cost alternative] + [Increment between them] The “defender” is the best alternative identified so far in the process, and “challenger” is the next higher-cost alternative to be evaluated. For a set of N mutually exclusive alternatives, (N - 1) “challenger/defender” comparisons must be made. Copyright Oxford University Press 2009 16

Example Given the alternatives below: Select the one best alternative if MARR = 8%. Use incremental rate of return analysis. 17

MARR = 8% Since the MARR is 8%, Alt. D may be eliminated, as the ROR is less than 8% Among the remaining alternatives A, B, and C, the two lower cost alternatives are A and B. (A - B) increment: PW of benefit = PW of cost (623 - 531)(P/A, i, 10) = (4, 000 - 3, 000) (P/A, i, 10) = 1, 000/92 = 10. 86 (C - B) increment: PW of benefit = PW of cost (1, 020 - 531)(P/A, i, 10) = (6, 000 - 3, 000) (P/A, i, 10) = 3, 000/489 = 6. 13 ∆ROR is greater than 8%. Therefore, choose the higher-cost alternative, Alt. C 18

Example 8 -1 Cost Benefit Life High Capacity $13, 400 $4000/year 5 years Low Capacity $10, 310 $3300/year 5 years Increment $3090 $700/year 5 years PW LOW= -$10, 310 + $3300(P/A, i, 5) PW HIGH= -$13, 400 + $4000(P/A, i, 5) IRRIncrement= 4. 3% IRRHigh IRRLow 19

Example 8 -1 – Continued In column B, input formula: PW LOW = –$10, 300 + $3300(P/A, i, 5) = –$10300 + pv(A 3, 5, – 3300) In column C, input formula: PW HIGH = –$13, 400 + $4000(P/A, i, 5) = –$13400 + pv(A 3, 5, – 4000) From a 3 to a 24, input interest from 0 to 0. 21 In column D, input formula for incremental cost PW HIGH–LOW: = C 3 – B 3 Then draw a line chart! Or use EXCEL function npv(i, value range) -10300+npv(A 3, $A$28: $A$32) -13400+npv(A 3, $A$36: $A$40)

Example 8 -1 – Continued Interest Rate PW LOW= -$10, 300 + $3300(P/A, i, 5) 0% ≤ i ≤ 4. 3% 4. 3 % ≤ i ≤ 18% ≤ i PW HIGH= -$13, 400 + $40000(P/A, i, 5) PWhigh-low = -$3090 + $700(P/A, i, 5) $8, 000 $6, 000 Diff IRR $40 High $20 $2, 000 ($4, 000) IRRIncrement= %4. 3 $4, 000 $0 0. 0% ($2, 000) Best Choice High Capacity Low Capacity Do Nothing 5. 0% 10. 0% Diff IRRLow $0 ($20)3. 5% 15. 0% ($40) 20. 0% High Low 4. 5% 25. 0% Diff 5. 5% 30. 0% ($60) ($80) 21

Net Present Worth Example 8 -2 Rate MARR = 10% Machine X $200 Machine Y $700 Uniform Annual Benefit $95 $120 End-of-Useful-Life Salvage Value $50 $150 Initial Cost Useful Life, in Years 6 In column C, – 700 + pv(A 3, 12, – 120, – 150) 12 Machine X Machine Y 0% $840. 00 $890. 00 1. 322 752. 24 2 710. 89 687. 31 4 604. 26 519. 90 6 515. 57 380. 61 8 441. 26 263. 90 10 378. 56 165. 44 12 325. 30 81. 83 14 279. 77 10. 37 16 240. 58 -51. 08 18 206. 65 -104. 23 20 177. 10 -150. 47 In column B, – 200 + pv(A 2, 6, -95, -50) + pv(A 2, 6, 0, 200 – pv(A 2, 6, -95, -50)) 22

Net Present Worth Example 8 -2 Rate Machine Y For MARR ≤ 1. 3%, Machine Y is the right choice Machine Y 0% $840. 00 $890. 00 1. 322 752. 24 2 710. 89 687. 31 4 604. 26 519. 90 6 515. 57 380. 61 8 441. 26 263. 90 10 378. 56 165. 44 12 325. 30 81. 83 14 279. 77 10. 37 16 240. 58 -51. 08 18 206. 65 -104. 23 20 177. 10 -150. 47 ∆ IRRIncrement =1. 3% IRRY Machine X For MARR ≥ 1. 3%, Machine X is the right choice 23

Example 8 -3 Consider the three mutually exclusive alternatives: Initial Cost Uniform Annual Benefit A $2000 B $4000 C $5000 410 639 700 Each alternative has a 20 year life and no salvage value. If the MARR is 6%, which alternative should be selected? 24

Initial Cost Uniform Annual Benefit A $2000 B $4000 C $5000 410 639 700 ∆ IRRC-B= 2% If MARR ≥ 9. 6%, Choose Alt. A ∆ IRRB-A=9. 6% Alt. B Alt. A Alt. C If 9. 6% ≥ MARR≥ 2%, Choose Alt. B If 2% ≥ MARR ≥ 0%, Choose Alt. C Net Present Worth Graph of Alternatives A, B, and C. 25

∆ IRRC-B= 2% If MARR ≥ 9. 6%, Choose Alt. A ∆ IRRB-A=9. 6% Alt. B Alt. A Alt. C If 9. 6% ≥ MARR≥ 2%, Choose Alt. B If 2% ≥ MARR ≥ 0%, Choose Alt. C How to find the intersection points: NPW(C-B) = -$5000+$4000+($700 -$639)(P/A, i, 20) = 0 ∆IRR(C-B) = 2% NPW(B-A) = -$4000+$2000+($639 - $410)(P/A, i, 20) = 0 ∆IRR(B-A) = 9. 6% 26

Example 8 -4 Cost Life Brass $100, 000 4 IRRTitanium - Stainless= 6. 3% Stainless $175, 000 10 Titanium $300, 000 25 If 6. 3% ≥ MARR ≥ 0%, Choose Titanium If 15. 3% ≥ MARR≥ 6. 3%, Choose Stainless IRRStainless - Brass=15. 3% If MARR ≥ 15. 3%, Choose Brass 27

Example 8 -5 Incremental Analysis (with Do-Nothing option) Initial Cost Uniform Annual Benefit Useful Life, in years IRRY-Z=3. 5% Machine X $200 65 6 IRRZ-X=11% X Machine Y $700 110 12 If MARR≥ 23%, IRRX=23% Choose “Do-Nothing” Z Y Machine Z $425 100 8 If 23%≥MARR≥ 11%, Choose X If 11%≥MARR≥ 3. 5%, Choose Z If 3. 5%≥MARR≥ 0%, Choose Y 28

Example 8 -5 Incremental Analysis (without Do-Nothing option) Initial Cost Uniform Annual Benefit Useful Life, in years IRRY-Z=3. 5% Machine X $200 65 6 IRRZ-X=11% X Machine Y $700 110 12 If MARR≥ 11%, IRRX=23% Choose X Z Y Machine Z $425 100 8 If 11%≥MARR≥ 3. 5%, Choose Z If 3. 5%≥MARR≥ 0%, Choose Y 29

Example 8 -6 Incremental Analysis using Graphical Comparison A $4000 639 Initial Cost Uniform Annual Benefit B $2000 410 IRRC-A=2% IRRA-B=9. 6% C $6000 761 D $1000 117 E $9000 785 Life = 20 yrs IRRB=20% 30

Example 8 -6 Incremental Analysis using Graphical Comparison Initial Cost Uniform Annual Benefit A $4000 639 B $2000 410 C $6000 761 D $1000 117 E $9000 785 Calculating Incremental Interest ∆IRR(C-A) = $6000 -$4000 = ($761 - $639)(P/A, i, 20) = 2% ∆IRR(A-B) = $4000 -$2000 = ($ 639 - $410)(P/A, i, 20) = 9. 6% And to find where the NPW of B crosses the 0 axis IRR (B) = $2000 = $410(P/A, i, 20) = 20% 31

Example 8 -6 Incremental Analysis using Graphical Comparison A $4000 639 Initial Cost Uniform Annual Benefit B $2000 410 D $1000 117 E $9000 785 If MARR≥ 20%, Choose Do-Nothing IRRC-A=2% IRRA-B=9. 6% C $6000 761 IRRB=20% If 20%≥MARR≥ 9. 6%, Choose B If 9. 6%≥MARR≥ 2%, Choose A If 2%≥MARR≥ 0%, Choose C 32

Spreadsheet and Incremental Analysis Excel Functions Purpose Rate (n, A, -P, [F], [Type], [guess]) To find rate of return or incremental rate of return given n, P, and A IRR (range, [guess]) To find internal rate of return (or incremental rate of return) of a series of cash flow (or incremental cash flow) Excel Tools Purpose Goal Seek It varies the value in one specific cell until a formula that's dependent on that cell returns the wanted result. Solver adjusts the values in the changing cells to produce the result from the target cell formula. Constraints are applied to restrict the values Solver can use in the model. Solver 33

End of Chapter 8 34