1. Chapter 8 Choosing the Best Alternative

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

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

4. 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

5. 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

6. 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

7. 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

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

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

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

11. 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

12. 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.

13. 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

14. 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.

15. 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

16. 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

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

18. 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

19. 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

20. 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)

21. 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

22. 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))

23. 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

24. 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?

25. Uniform Annual Benefit 410 639 700C
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.

26. 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%

27. 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%

28. 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

29. 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

30. Example 8-6 Incremental Analysis using Graphical ComparisonB 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%

31. Example 8-6 Incremental Analysis using Graphical ComparisonB 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%

32. Example 8-6 Incremental Analysis using Graphical ComparisonB 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%

33. 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.

34. End of Chapter 8