ISU CCEE
CE 203CE 203
Rate of Return Rate of Return AnalysisAnalysis(EEA Chapter 7)(EEA Chapter 7)
ISU CCEE
• “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
Rate of Return AnalysisRate of Return AnalysisRate of Return AnalysisRate of Return Analysis
ISU CCEE
• P = F (P/F, i, n) or P = A(P/A, i, n) EEA 5• A = P (A/F, i, n) EEA 6
Rate of Return Analysis, Rate of Return Analysis, RoRRoRRate of Return Analysis, Rate of Return Analysis, RoRRoR
EEA 7: i?for benefits = costs
ISU CCEE
• 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)
Rate of Return AnalysisRate of Return AnalysisRate of Return AnalysisRate of Return Analysis
ISU CCEE
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
In-class example 7-1In-class example 7-1In-class example 7-1In-class example 7-1
ISU CCEE
Use plotting:
In-class example 7-1In-class example 7-1In-class example 7-1In-class example 7-1
ISU CCEE
• 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?)
Rate of Return AnalysisRate of Return AnalysisRate of Return AnalysisRate of Return Analysis
ISU CCEE
• NPW = $5000• EUAW = $800• RoR = 8%
What is easier to What is easier to understand?understand?What is easier to What is easier to understand?understand?
ISU CCEE
• Investment: subsequent inflow >
initial amount
• Borrowing: subsequent outflow >
initial amount
• Usually (but not always) investigate initial cash flow– Investment if negative– Borrowing if positive
Investment vs. Borrowing Investment vs. Borrowing SituationSituationInvestment vs. Borrowing Investment vs. Borrowing SituationSituation
ISU CCEE
Investment vs. Borrowing Investment vs. Borrowing Example Example Investment vs. Borrowing Investment vs. Borrowing Example Example
Year Cash Flow #1 Cash Flow #2
0 -$5,000 $5,000
1 $1,000 -$1,000
2 $3,000 -$3,000
3 $2,000 -$2,000
Sum = $1,000 Investment
Sum = - $1,000 Borrowing
Investment Borrowing
ISU CCEE
• 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.)
Minimum Attractive Rate of Minimum Attractive Rate of ReturnReturnMinimum Attractive Rate of Minimum Attractive Rate of ReturnReturn
ISU CCEE
• 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 (IRR) on the difference between the alternatives and compare to MARR If IRR > MARR, choose higher-cost alternative If IRR < MARR, choose lower-cost alternative
Rate of Return AnalysisRate of Return AnalysisRate of Return AnalysisRate of Return Analysis
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Alternative #1 Alternative #2
0 - $5,000 - $5,000
1 $4,500 $500
2 $1,400 $5,700
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
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Alternative #1 Alternative #2
0 - $5,000 - $5,000
1 $4,500 $500
2 $1,400 $5,700
Total C. F. +$900 + $1200
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 IRR
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Alt. #1 Alt. #2 Alt. #2 – Alt. #1
0 - $5,000 - $5,000 $0
1 $4,500 $500 - $4000
2 $1,400 $5,700 +$4,300
Total C. F. +$900 + $1200 + $300
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
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Alt. #1 Alt. #2 Alt. #2 – Alt. #1
0 - $5,000 - $5,000 $0
1 $4,500 $500 - $4000
2 $1,400 $5,700 +$4,300
Total C. F. +$900 + $1200 + $300
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) -2
For this simple case, i = 300/4000 = .075 = 7.5% = IRR
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Alt. #1 Alt. #2 Alt. #2 – Alt. #1
0 - $5,000 - $5,000 $0
1 $4,500 $500 - $4000
2 $1,400 $5,700 +$4,300
Total C. F. +$900 + $1200 + $300
Payback alternatives for an initial investment of $5000 (sum of cash flows both > $0). MARR = 6%.
Since IRR = 7.5% > MARR = 6%, choose alternative #2
ISU CCEE
In-class Example 7-2In-class Example 7-2In-class Example 7-2In-class Example 7-2
Year Action
0 Invest $5,000
1 Receive $4,500 and invest it at 6% (MARR)
2 Receive $1,400 + $4,500 (1 + .06) = $6,170
Or, to look at it another way:Alternative #1
Year Action
0 Invest $5,000
1 Receive $500 and invest it at 6% (MARR)
2 Receive $5,700 + $500 (1 + .06) = $6,230
Alternative #2
ISU CCEE
In-class Example 7-2 with MARR In-class Example 7-2 with MARR = 9%= 9%In-class Example 7-2 with MARR In-class Example 7-2 with MARR = 9%= 9%
Year Alternative #1 Alternative #2
0 - $5,000 - $5,000
1 $4,500 $500
2 $1,400 $5,700
Suppose MARR = 9% for payback alternatives for an initial investment of $5000:
Since DRoR = 7.5% < MARR = 9%, choose alternative #1
ISU CCEE
In-class Example 7-2 with MARR In-class Example 7-2 with MARR = 9%= 9%In-class Example 7-2 with MARR In-class Example 7-2 with MARR = 9%= 9%
Year Action
0 Invest $5,000
1 Receive $4,500 and invest it at 9% (MARR)
2 Receive $1,400 + $4,500 (1 + .09) = $6,305
Alternative #1
Year Action
0 Invest $5,000
1 Receive $500 and invest it at 9% (MARR)
2 Receive $5,700 + $500 (1 + .09) = $6,245
Alternative #2
ISU CCEE
In-class Example 7-2 but we are the In-class Example 7-2 but we are the borrowerborrowerIn-class Example 7-2 but we are the In-class Example 7-2 but we are the borrowerborrower
Year Alternative #1 Alternative #2
0 - $5,000 - $5,000
1 $4,500 $500
2 $1,400 $5,700
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.
ISU CCEE
In-class Example 7-3In-class Example 7-3In-class Example 7-3In-class Example 7-3
Year Cash flow
0 $1,020
1 - $2,000
2 $500
3 $500
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.
ISU CCEE
In-class Example 7-3In-class Example 7-3In-class Example 7-3In-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 vs. Interest Rate
-15
-10
-5
0
5
10
15
20
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Interest Rate
Net
Pre
sen
t W
ort
h
NPW = 0 at i values of 5.24% and 27.4%
Two answers!
ISU CCEE
Multiple values for ROR Multiple values for ROR possible!possible!Multiple values for ROR Multiple values for ROR possible!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)
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