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**Chapter 8 Choosing the Best Alternative**

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**Chapter Outline Incremental Analysis**

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

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**Learning Objectives Define Incremental Analysis**

Apply Graphical Technique in Solving Problems with Mutually Exclusive Alternatives Use Spreadsheets in Incremental Analysis

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**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

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

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**QUESTION CONTINUES -8.85 YEAR CASH FLOW ($500) 1 $100 2 $150 3 $200 4**

($500) 1 $100 2 $150 3 $200 4 $250 QUESTION CONTINUES -8.85

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**QUESTION CONTINUES -8.85 YEAR CASH FLOW ($500) 1 $100 2 $150 3 $200 4**

($500) 1 $100 2 $150 3 $200 4 $250 QUESTION CONTINUES -8.85

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INTERPOLATION 5% 15% X% 30.95 -8.85 5-X 10 39.80

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INTERPOLATION 5% 15% X% 30.95 -8.85 5-X 10 39.80

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EXCEL solution IRR = irr(a1:a5) = 12.83%

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**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

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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.

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**Two -Alternative Situations**

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

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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.

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**Choose the higher-cost alternative Choose the lower-cost alternative**

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

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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

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

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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 ( )(P/A, i, 10) = (4, ,000) (P/A, i, 10) = 1,000/92 = 10.86 (C - B) increment: PW of benefit = PW of cost (1, )(P/A, i, 10) = (6, ,000) (P/A, i, 10) = 3,000/489 = 6.13 ∆ROR is greater than 8%. Therefore, choose the higher-cost alternative, Alt. C

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**Example 8-1 High Capacity Low Capacity Increment Cost $13,400 $10,310**

$3090 Benefit $4000/year $3300/year $700/year Life 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

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**Example 8-1 – Continued In column B, input formula:**

PW LOW = –$10,300 + $3300(P/A,i,5) = –$ pv(A3, 5, –3300) In column C, input formula: PW HIGH = –$13,400 + $4000(P/A,i,5) = –$ pv(A3, 5, –4000) From a3 to a24, input interest from 0 to 0.21 In column D, input formula for incremental cost PW HIGH–LOW: = C3 – B3 Then draw a line chart! Or use EXCEL function npv(i, value range) npv(A3, $A$28:$A$32) npv(A3, $A$36:$A$40)

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**Example 8-1 – Continued PW LOW= -$10,300 + $3300(P/A,i,5)**

Interest Rate Best Choice 0% ≤ i ≤ 4.3% High Capacity 4.3 % ≤ i ≤ 18% Low Capacity 18% ≤ i Do Nothing PW HIGH= -$13,400 + $40000(P/A,i,5) PWhigh-low = -$ $700(P/A, i, 5) IRRIncrement= %4.3 IRRIncrement= %4.3 IRRHigh IRRLow

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**Example 8-2 Net Present Worth Rate 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 20 177.10 MARR = 10% Machine X Machine Y Initial Cost $200 $700 Uniform Annual Benefit $95 $120 End-of-Useful-Life Salvage Value $50 $150 Useful Life, in Years 6 12 In column C, –700 + pv(A3, 12, –120, –150) In column B, –200 + pv(A2, 6, -95, -50) + pv(A2,6, 0, 200 – pv(A2, 6, -95, -50))

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**Example 8-2 Net Present Worth Rate 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 20 177.10 ∆ IRRIncrement =1.3% IRRY Machine X Machine Y For MARR ≤ 1.3%, Machine Y is the right choice For MARR ≥1.3%, Machine X is the right choice

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**Example 8-3 Consider the three mutually exclusive alternatives: A B C**

Initial Cost $2000 $4000 $5000 Uniform Annual Benefit 410 639 700 Each alternative has a 20 year life and no salvage value. If the MARR is 6%, which alternative should be selected?

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**Uniform Annual Benefit 410 639 700**

C Initial Cost $2000 $4000 $5000 Uniform Annual Benefit 410 639 700 ∆ IRRC-B= 2% If MARR ≥ 9.6%, Choose Alt. A If 9.6% ≥ MARR≥2%, Choose Alt. B If 2% ≥ MARR ≥ 0%, Choose Alt. C ∆ IRRB-A=9.6% Alt. B Alt. A Alt. C Net Present Worth Graph of Alternatives A, B, and C.

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**If MARR ≥ 9.6%, Choose Alt. A If 9.6% ≥ MARR≥2%, Choose Alt. B**

∆ IRRC-B= 2% If MARR ≥ 9.6%, Choose Alt. A If 9.6% ≥ MARR≥2%, Choose Alt. B If 2% ≥ MARR ≥ 0%, Choose Alt. C ∆ IRRB-A=9.6% Alt. B Alt. A 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%

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**Example 8-4 Brass Stainless Titanium Cost $100,000 $175,000 $300,000**

Life 4 10 25 If 6.3% ≥ MARR ≥ 0%, Choose Titanium If 15.3% ≥ MARR≥ 6.3%, Choose Stainless If MARR ≥ 15.3%, Choose Brass IRRTitanium - Stainless= 6.3% IRRStainless - Brass=15.3%

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**Example 8-5 Incremental Analysis (with Do-Nothing option)**

Machine X Machine Y Machine Z Initial Cost $200 $700 $425 Uniform Annual Benefit 65 110 100 Useful Life, in years 6 12 8 IRRY-Z=3.5% If MARR≥23%, Choose “Do-Nothing” If 23%≥MARR≥11%, Choose X If 11%≥MARR≥3.5%, Choose Z If 3.5%≥MARR≥0%, Choose Y IRRZ-X=11% IRRX=23% X Z Y

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**Example 8-5 Incremental Analysis (without Do-Nothing option)**

Machine X Machine Y Machine Z Initial Cost $200 $700 $425 Uniform Annual Benefit 65 110 100 Useful Life, in years 6 12 8 IRRY-Z=3.5% If MARR≥11%, Choose X If 11%≥MARR≥3.5%, Choose Z If 3.5%≥MARR≥0%, Choose Y IRRZ-X=11% IRRX=23% X Z Y

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**Example 8-6 Incremental Analysis using Graphical Comparison**

B C D E Initial Cost $4000 $2000 $6000 $1000 $9000 Uniform Annual Benefit 639 410 761 117 785 IRRC-A=2% Life = 20 yrs IRRA-B=9.6% IRRB=20%

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**Example 8-6 Incremental Analysis using Graphical Comparison**

B C D E Initial Cost $4000 $2000 $6000 $1000 $9000 Uniform Annual Benefit 639 410 761 117 785 Calculating Incremental Interest ∆IRR(C-A) = $6000-$4000 = ($761 - $639)(P/A, i, 20) = 2% ∆IRR(A-B) = $4000-$2000 = ($ $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%

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**Example 8-6 Incremental Analysis using Graphical Comparison**

B C D E Initial Cost $4000 $2000 $6000 $1000 $9000 Uniform Annual Benefit 639 410 761 117 785 If MARR≥20%, Choose Do-Nothing If 20%≥MARR≥9.6%, Choose B If 9.6%≥MARR≥2%, Choose A If 2%≥MARR≥0%, Choose C IRRC-A=2% IRRA-B=9.6% IRRB=20%

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**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 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.

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End of Chapter 8

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