Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf ·...
Transcript of Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf ·...
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Chapter 8:
Rate of Return Analysis: Multiple
Alternatives
Session 24, 25
Dr Abdelaziz Berrado
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Topics to Be Covered in Today’s Lecture
Section8.1: Why Incremental Analysis is
Necessary?
Section8.2: Calculation of Incremental Cash Flows
For Analysis
Section8.3: Interpretation of Rate of Return on the
Extra Investment
Section8.4: ROR Evaluation by PW: Incremental
and Breakeven
Section8.5: ROR Evaluation Using AW
Section8.6: Incremental Analysis for the Analysis
of Multiple Alternatives
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Section 8.1: Why Incremental Analysis is Necessary?
• Assume we have two or more mutually exclusive alternatives
•Objective: Which, if any of the alternatives is preferred?
•Prior Chapters: Use the PW or AW approach
•This chapter: We apply the ROR approach
•Present Worth: Equal service lives must apply
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8.1 Ranking Inconsistency
• For some problems, PW and ROR may rank the same problems differently. Why?
•PW assumes reinvestment at the MARR or discount rate.
•ROR assumes reinvestment at the i* or i’ rate
•Two different reinvestment rate assumptions apply
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8.1 An Example – Shows Ranking
Inconsistency Problem
•Two Investments A and B
•Discount rate = 10%
•Each investment requires $100 at t = 0
•A is a 1-year investment
•B is a 5- year investment
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8.1 Two Projects; A and B
• A
$100
0 1 2 3 4 5
$120
• B
$100
0 1 2 3 4 5
$201.14
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8.1 Example Problem
• i*A = 0.20 = 20%
• i*B = 0.15 = 15%
•PWA(10%) = +$9.09
•PBB(10%) = +24.89
•Using ROR, A is superior to B
•Using PW, B is superior to A
•Inconsistent Rankings!
• Using ROR Ranking
•A is superior to B (20% > 15%)
•Using a PW(10%) approach
•B is superior to A ($24.89 > $9.09)
•The two methods do not rank the same?
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8.1 Look at A and assume Reinvestment
forward to t = 5
• Reinvest the $120 out to t = 5
$100
0 1 2 3 4 5
$120
•Assume the +$120 can be reinvested forward at the firm’s MARR rate of 10%/year out to the end of year 5.
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8.1 Look at A and assume Reinvestment
forward to t = 5
• Find F5 for Alt. A
$100
0 1 2 3 4 5
$120
F5 = ?
•F5 = 120(F/P,10%,4)=$175.69
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8.1 ROR of A given reinvestment
• -100 +120(F/P,10%,4)(P/F,i*A,5) = 0
•Solving for i*A
•(P/F, i*A,5) = 0.569
• i*A/reinvestment @10% = 0.1193
•i*A/c=10% = 11.93%
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8.1 Now Compare A to B• Compare revised A with reinvestment at 10% to B
•i*A/C=10% = 11.93%
•i*B = 15% as before
•ROR Rankings:
•B is superior to A (15% > 11.93%)
•Now, PW(10%) and ROR with the reinvestment imposed on the 1-year project rank consistently.•B is superior to A with both methods
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8.1 Ranking Inconsistency…
• Occurs between ROR and PW because
•Both methods have different reinvestment rate assumptions
•Two different cash flows may not generate funds at the same rate
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Section 8.2: ROR for Mutually
Exclusive Projects
•Given Two or more alternatives
•Rank the investments based upon their initial time t = 0 investment requirements
•Summarize the investments in a tabular format
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8.2 Tabular Format
$$N
………
$$2
$$1
$$0
B - AAlt. BAlt. At
Find the ROR of this investment which is
(B – A)
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8.2 Ranking Rules
• Given two or more Mutually Exclusive investments
•Select the first investment to be the one with the lowest time t = 0 investment amount.
•The next investment is to be the one with the largest investment at time t = 0
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8.2 Example
• Two Investments: A and B
• A costs $30,000 at time t = 0
• B costs $50,000 at time t = 0
•MARR = 10%
•Life is 4 years
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8.2 Example: A and B• For this problem, A is superior to B based on PW and on ROR!
•A is ranked first;
•B is ranked second
•Both alternatives have
a PW > 0 and have i*’s
> MARR.
•Both are feasible
alternatives initially.
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8.2 Form the Difference (B – A)
• For mutually exclusive alternatives…
•One should focus on the differences between the alternatives
•“Differences” are illustrated best by forming what is called the incremental investment (B-A)
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8.2 Incremental Investment
Lowest FirstCost
investment
Next Highest first
Costinvestment
The Incrementalinvestment
A B (B-A)
Find the ROR of this investment
=
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8.2 Incremental Investment
B - A
The incremental Investment
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8.2 The logic….
• One would go with investment A initially because it is the least expensive alternative at time t = 0
•And it’s present worth is > 0.
•So, A is a feasible alternative to start
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8.2 Explaining the Incremental Investment
• Now, is it worth it to the firm to consider investing (-$50,000 – (-$30,000) =
-$20,000 to get the cash flows indicated in the (B-A) cash flow series?
B-A
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8.2 Explaining….continued
A B (B-A)
The investment (B-A) represents the year-by-year difference between A and B
(B-A) is “additional” investment to move from
investing in A and moving on to invest in B.
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8.2 The Incremental Investment
• Investing $20,000 at time t = 0 results in the following incremental investment
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8.2 Is it worth it?
• Now the question is….
•Is it worth spending an additional $20,000 to move from investment A to investment B?
•Answer: Compute the ROR or PW of the incremental investment to see!
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8.2 Analysis
• For this problem, NPV(10%) < 0
• and, no ROR could be found!
•The incremental investment shows a negative PW and no ROR is found•Thus, the increment is rejected.•Moving from A to B is not economically worth it•Stay with A!
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8.2 Another Example
•Cash Flows are shown on the next slide
• Two alternatives
•Semiautomatic machine vs.
•Automatic machine
•Assume a 6 year life for analysis
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8.2 Example
15,000-85,000-100,0005
21,000-74,000-95,0006
15,000-85,000-100,0004
50,000-85,000-135,0003
15,000-85,000-100,0002
15,000-85,000-100,0001
-$50,000-$90,000-$40,0000
(B-A)Auto
B
SemiAuto
A
Year
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8.2 Analysis
• Computed PW @ 18% shows that B has the lowest PW cost and would be preferred to A
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8.2 Incremental Cash Flow
21,000
15,000
15,000
50,000
15,000
15,000
-$50,000
(B-A)
6
5
4
3
2
1
0
•Question?
•Is it worth spending an additional $50,000 in the automatic machine in order to receive the incremental savings shown to the left?
•Compute the ROR of the incremental Cash Flow
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8.2 Incremental ROR = 35.95%
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8.2: NPV Plot of A and B
• A is equivalent to B @ Incremental ROR rate of 35.95%
NPV PLOT-INC. C.F.
-800000.00
-700000.00
-600000.00
-500000.00
-400000.00
-300000.00
-200000.00
-100000.00
0.00
0.0
0
0.1
0
0.2
0
0.3
0
0.4
0
0.5
0
0.6
0
0.7
0
0.8
0
0.9
0
1.0
0
1.1
0
1.2
0
1.3
0
1.4
0
1.5
0
1.6
0
1.7
0
1.8
0
1.9
0
2.0
0
Disc. Rates
NP
V(i
%)
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8.2: i* (B-A) = 35.95%
• The incremental i*(B-A) is greater than the firm’s
discount rate of 18%
•Since i*(B-A) > MARR, accept the increment and
go with Alternative B.
•Same results as PW(18%) shows
•B is clearly the winner
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Section 8.3 Interpretations of ROR
• The i*incremental is the ROR of the additional or
incremental investment required to move from
one project to the next most costly project
•If the i*incremental value is < MARR the increment
is not worth it. Go with to lower investment cash
flow
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8.3 Multiple Alternatives
• If Cost-Revenue Problem…
•Calculate the computed i*’s for each alternative in the set
•Discard those alternatives whose i* value is less than the MARR – they would lose anyway!
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8.3 Independent Projects
• If dealing with independent projects, one does not compute incremental investments among the candidate projects
•Rule: Accept all projects whose ROR > MARR and stay within any budget limitations
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8.3 The i* (B-A) value
• Given two mutually exclusive alternatives,
A and B.
•The i* (B-A) value also represents the
interest rate at which the two alternatives
are economically equivalent.
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8.4 ROR Using PW: Incremental
and Breakeven– Mutually Exclusive Case
• Selection between multiple mutually exclusive alternatives by the IROR method.
•If unequal lives – either establish a common project life or,
•Apply the LCM of life approach found in chapter 5
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8.4 PW Approach – Mutually
Exclusive Case
• Order (rank) the alternatives by their initial time t = 0 investment cost
• Start with the smaller investment alternative – refer to it as “A”
•The next highest investment cost is called “B”
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8.4 PW Approach – Mutually
Exclusive Case
• Compute the incremental cash flow (B-A)
• Given the MARR find the PW of (B-A) investment
• If PW(MARR) of (B-A) is >0; accept the increment – go with the higher investment cost alternative.
•Else, reject the increment and go with the lower investment cost option
![Page 41: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/41.jpg)
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8.4 ROR Case – Unique i* (B-A)
• Compose the incremental Cash Flow
•Examine that cash flow for sign changes and
apply the Norstrom test (from Chapter 7)
•If a unique i* (B-A) is indicated – solve for it
and compare it to the MARR
•If i* (B-A) > MARR, accept the increment else
reject
![Page 42: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/42.jpg)
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8.4 Example 8.3 Bell/GTE
•10 year project (merger)
•New equipment is required
•Two vendors
•MARR = 15%
•Which vendor should be selected
•Cost or Service Problem
![Page 43: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/43.jpg)
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8.4 Setup A vs. BPW(15%) shows that A has the
lowest PW Cost and should win!
•Note: No ROR for A (all negative signs) and B’s cannot be determined!
![Page 44: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/44.jpg)
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8.4 PW analysis
• We could stop because the PW(15%) has signaled that A is the winner!
•Lowest PW cost
•Proceed with a ROR analysis BUT….
•ROR must be performed on the incremental investment
![Page 45: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/45.jpg)
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8.4 Incremental Cash Flow Fig. 8-2
![Page 46: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/46.jpg)
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8.4 Inc. Cash Flow Results
Inc. PV(15%) and is
negative. Thus, reject the increment and go with A!
i* (B-A) is less
than the
MARR of
15%.
Reject
increment
and go with
A!
![Page 47: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/47.jpg)
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8.4 So-called Breakeven ROR
• Recall, the incremental i*(B-A) is the interest rate at which the two alternatives are economically equivalent.
•This special interest rate is called:
•Breakeven Interest Rate
•Fisherian Intersection Rate
![Page 48: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/48.jpg)
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8.4 Breakeven Rate Illustrated
• For Example 8.3 the NPV Plot is: NPV PLOT-INC. C.F.
-50000.00
-45000.00
-40000.00
-35000.00
-30000.00
-25000.00
-20000.00
-15000.00
-10000.00
-5000.00
0.00
0.0
0
0.1
0
0.2
0
0.3
0
0.4
0
0.5
0
0.6
0
0.7
0
0.8
0
0.9
0
1.0
0
1.1
0
1.2
0
1.3
0
1.4
0
1.5
0
1.6
0
1.7
0
1.8
0
1.9
0
2.0
0
Disc. Rates
NP
V(i
%)
i*(B-A) rate;Alternatives are identical at this rate.
![Page 49: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/49.jpg)
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8.4 Conclusions i* (B-A) = 12.65%
• For MARR < 12.65% extra investment is justified. Go with B
•For MARR > 12.65%, the extra investment is not justified: Go with A
•If MARR = 12.65%, both options are economically equivalent.
Recall: MARR = 15%
![Page 50: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/50.jpg)
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8.5 ROR Evaluation Using Annual
Worth based relation
• ROR approach requires comparison over an equal-service life
•When the lives are equal or unequal (Must use LCM), set up the AW relationship for the cash flows of each alternative
•Then solve 0 = AWB – AWA for the i* value
![Page 51: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/51.jpg)
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8.5 ROR Using Annual Worth
• See Example 8.5
•Manual approach
•It is best to avoid this approach and stay with either the PW or ROR over the total life of the project or the LCM life
![Page 52: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/52.jpg)
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8.6 Incremental ROR Analysis of Multiple,
Mutually Exclusive Alternatives: Criteria
•Select the one alternative that:
•Requires the largest investment
•And indicates that the extra investment over another acceptable alternative is justified
![Page 53: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/53.jpg)
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8.6 Comparing Alternatives
• A given alternative should not be compared with one alternative for which the incremental investment is not justified
•If a given alternative looses out in a comparison, that alternative is dropped from further consideration.
![Page 54: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/54.jpg)
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8.6 Ranking Rules - Ordering
1. Order the alternatives from smallest to largest initial investment
2. Compute the cash flows for each alternative (assume or create equal lives)
3. If the alternatives are revenue-cost alternatives do the following…
![Page 55: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/55.jpg)
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8.6 Revenue-Cost Problems
4. Compute the i* value for all alternatives in the considered set.
•If any alternative has an i* < MARR drop it from further consideration
•The candidate set will be those alternatives with computed i* values > MARR.
•Call this the FEASIBLE set
![Page 56: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/56.jpg)
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8.6 Revenue-Cost - Approach
• Calculate i* for the first alternative
•The first alternative is called the DEFENDER
•The second (next higher investment cost) alternative is called the CHALLENGER
•Compute the incremental cash flow as
•(Challenger – Defender)
![Page 57: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/57.jpg)
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8.6 Revenue-Cost
4. Compute i*Challenger – Defender
•If i*Challenger – Defender > MARR drop the defender and the challenger wins the current round.
5. If i* Challenger – Defender < MARR, drop the challenger and the defender moves on to the next comparison round
![Page 58: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/58.jpg)
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8.6 Revenue-Cost
• At each round, a winner is determined
•Either be the current Defender or the current challenger
•The winner of a given round moves to the next round and becomes the current DEFENDER and is compared to the next challenger
![Page 59: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/59.jpg)
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8.6 Revenue-Cost
6. This process continues until there are no more challengers remaining.
•The alternative that remains after all alternatives have been evaluated is the final winner.
![Page 60: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/60.jpg)
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8.6 Potential Problems
• See Example 8.6 for the potential problems that can occur using ROR
•Pitfalls often exist when one uses the ROR approach for the analysis of alternatives
![Page 61: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/61.jpg)
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8.6 Cost Problems
• Remember
•Cost problems do not have computed ROR’ssince there are more cost amounts than revenue amounts (salvage values may exist)
•Thus there are no feasible i*’s for each alternative
![Page 62: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/62.jpg)
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8.6 Cost Problems - Rules
• Rank the alternatives according to their investment requirements (low to high)
•For the first round compare:
•Challenger – Defender Cash Flow
•Compute i*Challenger – Defender
•If i* Challenger – Defender > MARR, Challenger wins; else Defender wins
![Page 63: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/63.jpg)
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8.6 Cost Problems
• The current winner now becomes the defender for the next round.
•Compare the current defender to the next challenger and compute i* Challenger – Defender
•The winner becomes the current champion and moves to the next round as the defender
•Repeat until all alternatives have been compared.
![Page 64: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/64.jpg)
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Chapter 8 Summary
•PW and AW methods are preferred methods for evaluating alternatives
•ROR can be used but care must be taken
•If ROR, must perform an incremental analysis
•Two at a time (paired comparison) is required
![Page 65: Chapter 8: Rate of Return Analysis: Multiple AlternativesA.Berrado/EGR2302/EGR2302_Ch08.pdf · EGR2302-Engineering Economics Al Akhawayn University 1 Chapter 8: Rate of Return Analysis:](https://reader036.fdocuments.us/reader036/viewer/2022062402/5afbcdb67f8b9aa34d8b4bb2/html5/thumbnails/65.jpg)
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Assignments and Announcements
� Homework7 (Chapter8 due April 26th)
� Assignments due at the beginning of next
class:
� Online Quizzes for chapter 8 Due
� Read Chapter 14