Solutions to end-of–chapter problems
Engineering Economy, 8th edition
Leland Blank and Anthony Tarquin
Chapter 7
Rate of Return Analysis: One Project
Understanding ROR
7.1 (a) Highest possible is infinity
7.2 Total amount owed = principal x interest of 10% per year on principal for 5 years
Loan balance = Total amount owed – total amount paid
7.3 (a) Annual payment = principal/# of periods + interest per year
(b) A = 10,000(A/P,10%,4)
7.4 (a) Unrecovered balance of principal before payment, year 1 = $60,000,000
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Recovered amount, year 1 = payment – interest due
= 15,027,600 – 4,800,000
= $10,227,600
Unrecovered balance of principal after payment, year 1 = 60,000,000 – 10,227,600
= $49,772,400
7.5 Monthly payment = 100,000(A/P,0.5%,360)
Balloon payment = 100,000(F/P,0.5%,60) – 600(F/A,0.5%,60)
7.6 Factors: Annual payment = 6,000,000(A/P,10%,10)
Determination of ROR
7.7 (a) Factors and interpolation:
(b) Spreadsheet: If cash flows are entered into cells B1:B11
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of McGraw–Hill Education.
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= RATE(10,225000,-650000,70000,,20%) displays 32.8%
7.8 Interpolation:
Spreadsheet: Must use IRR function; enter cash flows into cells B1:B9.
7.9 Factors: Move all cash flows to year 1
Spreadsheet: Enter amounts for years 0 to 4 in cells B2:B6 or years 1 to 4 in cells B3:B6
7.10 0 = -50,000(8) + [(10(2500) + 25(650) +70(1200)](P/A,i*%,4)
Solve for i* by interpolation in interest tables or spreadsheet
7.11 0 = -4000 – 300(P/A,i*,4)(P/F,i*,1) – 100(P/A,i*,3)(P/F,i*,5) + 12,000(P/F,i*,8)
Solve for i* by trial and error or spreadsheet
7.12 (a) 0 = –650,000 + 105,000(P/A,i*,5) + 50,000(P/F,i*,5)
(b) Let A = annual savings. Solve by factor or Goal Seek
0 = -650,000 + A(P/A,15%,5) + 50,000(P/F,15%,5)
A = $186,490 (Goal Seek)
7.13 There are no tables for negative interest rates. Write the equation for i < 0% values
7.15 0 = –30,000 + (27,000 – 18,000)(P/A,i*,5) + 4000(P/F,i*,5)
7.17 Factors: 0 = -130,000 – 49,000(P/A,i*,8) + 78,000(P/A,i*,8) + 1000(P/G,i*,8)
Year Cash flow, $
0–130,000
129,000
230,000
331,000
432,000
533,000
735,000
859,000
i* via IRR 19.2%
7.18 (a) In $ million units
(b) Cost of guardrail = 72,000(113) = $8.136 million
Spreadsheet: = RATE(10,1100000,-72000*113) displays i* = 5.9%
7.19 Pi* = A
7.22 0 = -950,000 + [450,000(P/A,i*,5) + 50,000(P/G,i*,5)](P/F,i*,10)
Multiple ROR Values
7.23 Reinvestment rate assumes that any net positive cash inflows to a project make a return of a
7.24 Descartes’ rule uses net cash flows while Norstrom’s criterion is based on cumulative cash
flows.
7.27 (a) Net cash flow: $-1500, $260, $250, $375, $90, and $230
Cum. cash flow: $-1500, $-1240, $-990, $-615, $-525, and $-295
7.29 Entering net cash flows into cells B2:B5 results in = IRR(B2:B5), which displays
i* = 166.0%
7.30 Tabulate net cash flows and cumulative cash flows.
Quarter Expenses Revenue Net Cash Flow Cumulative
0 -20 0 -20 -20
(a) From net cash flow column, there are two possible i* values
7.31 (a) Two sign changes; maximum number of i* values is two.
(b) Cumulative cash flow series changes sign once. There is one, nonnegative i* value.
Cumulative
Year Net Cash Flow, $ Cash Flow, $
4 -1000 +7000
(c) 0 = –40,000 + 32,000(P/F,i*,1) + 18,000(P/F,i*,2) – 2000(P/F,i*,3) – 1000(P/F,i*,4)
7.32 Rule of signs: three possible i* values.
Cumulative cash flow signs: one, positive i* value.
Write PW rate of return equation (in $1000) to find i*
Solve for i* by trial and error or spreadsheet
7.33 (a) Two sign changes in the NCF series; maximum of two real-number i* values.
Norstrom’s criterion indicates one positive value.
7.34 (a) Three sign changes in NCF series; three possible i* values. Norstrom’s
criterion also indicates that there may be more than one i* value.
7.35 Calculate net cash flows and cumulative cash flows.
Year Expenses, $ Savings, $ Net Cash Flow, $ Cumulative CF, $
0 -33,000 0 -33,000 -33,000
4 -13,000 12,000 -1000 +2000
(a) Four sign changes in net cash flow; up to four i* values.
-$20,000
$0
$20,000
$40,000
$60,000
$80,000
i* vs. PW
7.36 (a) One sign change in NCF and cumulative CF indicates only one i*.
Removing Multiple i* Values
7.37 (a) Rule of signs test: up to 4 values
Cumulative CF sign test: inconclusive since S0 > 0
7.38 First find net cash flow (NCF)
Year 0 1 2 3 4
Revenue, $ 0 25,000 19,000 4,000 28,000
Costs, $ -6,000 -30,000 -7,000 -6,000 -12,000
NCF, $ -6000 -5000 12,000 -2000 16,000
7.39 (a) Cash flow rule of signs: up to three rate of return values.
Cumulative CF test: inconclusive; S0 > 0; series changes signs multiple times
7.40 (a) ROIC method with ii = 30% per year
(b) MIRR method with ii = 30% per year and ib = 10% per year
PW0 = -4000(P/F,ib,2) – 3000(P/F,ib,3)
7.41 Use MIRR function or hand solution.
(a) By hand, with ib = 10% and ii = 15%
= $-10,757