Solutions to end–of–chapter problems
Engineering Economy, 8th edition
Leland Blank and Anthony Tarquin
Chapter 4
Nominal and Effective Interest Rates
Nominal and Effective Rates
4.1 (a) week (b) quarter (c) six months
4.8 (a) r/8–mths: (0.09/3)*2 = 6%
4.12 (a) Interest period = year; CP = month; m = 12
(b) ia = (1 + r/m)m -1
4.13 APY = (1 + 0.12/12)12 – 1
4.14 (a) First find effective monthly rate; then find APR with m = 3 mths/quarter
(b) APY = (1 + 0.138/12)12 – 1
= 14.71%
4.15 (a) Interest rate per month = 4%
(b) ia = (1 + 0.48/12)12 – 1
4.16 (a) 0.1664 = (1 + 0.16/m)m – 1
(b) = EFFECT(16%,n)
4.17 0.75 = (1 + r/12)12 – 1
or using Equation [4.4] with m = 12
Equivalence When PP ≥ CP
4.18 (a) Since n is in years, need effective i/year
(b) Since n is semiannual periods, need effective i/6–mths
(c) Since n is quarters, need i/quarter
4.19 (a) Find the effective annual rates
(b) Determine the effective i per for a 6–month payment period.
4.20 (a) P = 65,000,000(P/F,3%,12)
4.21 Compound: F = 20,000,000(F/P,1.5%,12)
4.22 (a) P = 10,000(P/F,2%,28)
4.23 P = 15,000,000(P/F,5%,6)
4.25 F = 3.9(F/P,0.5%,120) (in $ billion)
4.27 F = 18,000(F/P,12%,4) + 26,000(F/P,12%,3) + 42,000(F/P,12%,2)
4.29 P = 1.3(P/A,1%,28)(P/F,1%,2) (in $ million)
4.31 P = 51(100,000)(0.25)(P/A,0.5%,60)
Find factor value by interpolation, formula, or spreadsheet.
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By formula: A = 4,100,000(0.35641) = $1,461,281 per year
By spreadsheet function: = PMT(15.865%,4,4100000)
A = $1,461,291 per year
4.33 (a) Need effective i per PP of 6 months
i/6 months = (1 + 0.03)2 – 1= 0.0609
4.34 F = 100,000(F/A,0.25%,8)(F/P,0.25%,3)
4.35 PP = year; CP = month; need effective annual i
ia = (1 + 0.18/12)12 – 1
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By spreadsheet, function = –PMT(19.562%,5,,FV(19.562%,3,350000)) displays
A = $171,975 per year
4.36 i/week = 0.25%
4.37 F = (14.99 – 6.99)(F/A,1%,24)
4.38 PP = 6 months; CP = quarterly; need effective i per PP
Find answer by interpolation, formula, or spreadsheet.
F = $249,776
4.39 (a) 3,000,000 = 200,000(P/A,1%,n)
NPER(1,200000,-3000000,0), respectively. Display is 16.3 months.
4.40 P = 80(P/A,3%,12) + 2(P/G,3%,12)
4.41 PP = quarter; CP = quarter; need i per quarter
4.42 PP = month; CP = month; use i = 1.5% per month; find P for g = 1%
4.43 PP = year; CP = 6 months; use effective annual i
(a) ia = (1 + 0.10/2)2 – 1 = 0.1025 (10.25% per year)
(b) Use the NPV and PMT functions at 10.25%
4.44 PP = month; CP = month; use i = 0.005 per month
(a) A = 80,000(A/P,0.5%,60) = 80,000(0.01933) = $1546.40 per month
Month
(1)
i per
month
(2)
Interest owed
(3)=(7 prior)(2)
Total owed
(4)=(7
prior)+(3)
Monthly
payment
(5)
Principal
reduction
Principal
remaining after
payment
(7)=(7 prior)–(6)
0
80,000.00
1
0.005
400.00
80,400.00
1546.40
78,853.60
2
0.005
394.27
79,247.87
1546.40
77,701.47 (b)
3
0.0035
271.96
77,973.43
1438.25 (d)
76,535,18
Equivalence When PP < CP
4.45 PP < CP. Move monthly deposits to end of 6-month compounding period; find F
4.46 Move deposit to end of annual compounding period; find F
4.47 Move cash flows to end of each quarter; find F
4.48 First find equivalent amount per 6-months and then divide by 6 to get monthly deposit
A per semiannual period = 80,000(A/F,6%,6)
4.49 Chemical cost/month = 11(30) = $330
4.50 Move withdrawals to beginning of semiannual periods; find F
Continuous Compounding
4.51 Formula: i = (e0.10 – 1) × 100 = 10.517%
Spreadsheet: = EFFECT(10%,10000)
4.52 r = 2% per month = 12% per 6 months
4.53 0.127 = er – 1 is the relation; r = ln (1 + i) provides the solution
4.54 r = 15% per year = 1.25% per month; need effective i per month
4.55 18% per year = 18/12 = 1.50% per month
4.56 (a) i = e0.02 – 1 = 2.02% per month
4.57 (a) i = e0.015 – 1 = 1.51% per month
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(b) Spreadsheet: = NPER(EFFECT(1.5%,10000),,-1,2) displays 46.2
4.58 (a) Set up F/P equation in months.
Varying Interest Rates
4.59 P = 150,000(P/F,12%,2)(P/F,10%,3)
4.60 Rates are 1%, increasing to 1.25% per month
F = 300,000(F/P,1%,4)(F/P,1.25%,8)
4.61 F value:
F = – 100(F/A,14%,5)(F/P,10%,3) – 160(F/A,10%,3)
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A value using relation for F:
-1409.40 = A(6.6101)(1.3310) +A(3.3100)
= A(12.1080)
A = $-116.40
4.62 First move cash flow in years 0-4 to year 4 at i = 12%; move the total to year 5 at i = 20%.
F4 = 5000(F/P,12%,4) + 6000(F/A,12%,4)
4.63 P = 5000(P/A,10%,3) + 7000(P/A,12%,2)(P/F,10%,3)
21,323 = A(P/A,10%,3) + A(P/A,12%,2)(P/F,10%,3)
Exercises for Spreadsheets
4.64 Solution of problem 4.19 using a spreadsheet.
4.65 Solution to Problem 4.44 using a spreadsheet.
4.66 (a) F = $9046; (b) F = $9139; Difference ≈ $92
4.67 Two spreadsheets follow the answers. You must write the functions.
(a) i/month = 0.06/12 = 0.005 (0.5%)
Parts (a) through (c) Parts (d) and (e)
Month
(payment #)
i per month Interest owed Total owed
Monthly
payment
Principal
reduction
Principal remaining
after payment
– 20,000.00
1.00 0.01 100.00 20,100.00 386.66 286.66 19,713.34
2.00
0.01
98.57 19,811.91 386.66 288.09 19,425.25
3.00 0.01
97.13
19,522.37 386.66 289.53 19,135.71
4.00 0.01
95.68
19,231.39
386.66
290.98 18,844.73
5.00 0.01 94.22 18,938.96
386.66
292.44
18,552.30
6.00 0.01 92.76 18,645.06 386.66 293.90 18,258.40
7.00 0.01 91.29 18,349.69 386.66 295.37 17,963.03
8.00 0.01 89.82 18,052.84 386.66 296.84 17,666.18
9.00 0.01 88.33 17,754.51 386.66 298.33 17,367.85
10.00
0.01 86.84 17,454.69 386.66 299.82 17,068.03
11.00
0.01 85.34 17,153.37 386.66 301.32 16,766.71
12.00 0.01
83.83
16,850.55 386.66 302.83 16,463.89
13.00 0.01
82.32
16,546.21
386.66
304.34 16,159.55
14.00 0.01 80.80 16,240.34
386.66
305.86 15,853.68
15.00 0.01 79.27 15,932.95 386.66 307.39 15,546.29
16.00 0.01 77.73 15,624.02 386.66 308.93 15,237.36
17.00 0.01 76.19 15,313.55 386.66 310.47 14,926.89
18.00 0.01 74.63 15,001.53 386.66 312.03 14,614.87
19.00 0.01 73.07 14,687.94 386.66 313.59 14,301.28
20.00
0.01 71.51 14,372.79 386.66 315.15 13,986.13
21.00 0.01
69.93 14,056.06 386.66 316.73 13,669.40
22.00 0.01 68.35 13,737.74 386.66 318.31 13,351.08
23.00 0.01 66.76 13,417.84 386.66 319.90 13,031.18
24.00 0.01 65.16 13,096.34 386.66 321.50 12,709.68
25.00 0.01 63.55 12,773.22 386.66 323.11 12,386.56
26.00 0.01 61.93 12,448.50 386.66 324.73 12,061.84
27.00 0.01 60.31 12,122.15 386.66 326.35 11,735.49
28.00 0.01 58.68 11,794.16 386.66 327.98 11,407.50
29.00 0.01 57.04 11,464.54 386.66 329.62 11,077.88
30.00 0.01 55.39
11,133.27
386.66
331.27 10,746.61
31.00 0.01 53.73 10,800.34 386.66 332.93 10,413.68
32.00
0.01 52.07 10,465.75 386.66 334.59 10,079.09
33.00 0.01 50.40 10,129.49 386.66 336.26 9,742.83
34.00 0.01 48.71 9,791.54 386.66 337.95 9,404.88
35.00 0.01 47.02 9,451.91 386.66 339.64
9,065.25
36.00 0.01 45.33 9,110.57 386.66 341.33 8,723.91
37.00 0.01 43.62 8,767.53 386.66 343.04 8,380.87
38.00 0.01 41.90
8,422.78 386.66 344.76 8,036.12
39.00 0.01 40.18
8,076.30
386.66 346.48 7,689.64
40.00 0.01 38.45 7,728.08
386.66
348.21 7,341.42
41.00 0.01 36.71 7,378.13 386.66 349.95
6,991.47
42.00
0.01
34.96 7,026.43 386.66 351.70 6,639.77
43.00 0.01 33.20 6,672.97 386.66 353.46 6,286.31
44.00 0.01 31.43 6,317.74
386.66
355.23 5,931.08
45.00 0.01 29.66 5,960.74 386.66 357.00 5,574.08
46.00
0.01 27.87 5,601.95 386.66 358.79 5,215.29
47.00 0.01
26.08 5,241.36 386.66 360.58 4,854.70
48.00 0.01 24.27
4,878.98
386.66 362.39 4,492.32
49.00 0.01 22.46 4,514.78 386.66 364.20 4,128.12
50.00 0.01 20.64 4,148.76 386.66 366.02
3,762.10
51.00 0.01
18.81 3,780.91 386.66 367.85 3,394.25
52.00 0.01 16.97 3,411.22 386.66 369.69 3,024.56
53.00 0.01 15.12 3,039.68
386.66 371.54 2,653.02
54.00 0.01 13.27 2,666.29 386.66 373.39 2,279.63
55.00 0.01 11.40 2,291.03 386.66 375.26 1,904.37
56.00 0.01 9.52 1,913.89 386.66 377.14 1,527.23
57.00 0.01 7.64 1,534.86
386.66 379.02 1,148.20
58.00 0.01 5.74 1,153.94 386.66 380.92 767.28
59.00 0.01 3.84 771.12 386.66 382.82 384.46
60.00
0.01 1.92 386.38 386.38 384.46 0.00
Totals 3,199.32 23,199.32
Month
(payment #)
i per month Interest owed Total owed
Monthly
payment
Principal
reduction
Principal remaining
after payment
– 20,000.00
10.005 100.00 20,100.00 386.66 286.66 19,713.34
20.005 98.57 19,811.91 386.66 288.09 19,425.25
30.005 97.13 19,522.37 386.66 289.53 19,135.71
40.005 95.68 19,231.39 386.66 290.98 18,844.73
50.005 94.22 18,938.96 386.66 292.44 18,552.30
60.005 92.76 18,645.06 386.66 293.90 18,258.40
70.005 91.29 18,349.69 386.66 295.37 17,963.03
80.005 89.82 18,052.84 386.66 296.84 17,666.18
90.005 88.33 17,754.51 386.66 298.33 17,367.85
10 0.005 86.84 17,454.69 386.66 299.82 17,068.03
11 0.005 85.34 17,153.37 386.66 301.32 16,766.71
12 0.005 83.83 16,850.55 386.66 302.83 16,463.89
13 0.005 82.32 16,546.21 386.66 304.34 16,159.55
14 0.005 80.80 16,240.34 386.66 305.86 15,853.68
15 0.005 79.27 15,932.95 386.66 307.39 15,546.29
16 0.005 77.73 15,624.02 386.66 308.93 15,237.36
17 0.005 76.19 15,313.55 386.66 310.47 14,926.89
18 0.005 74.63 15,001.53 386.66 312.03 14,614.87
19 0.005 73.07 14,687.94 386.66 313.59 14,301.28
20 0.005 71.51 14,372.79 386.66 315.15 13,986.13
21 0.005 69.93 14,056.06 386.66 316.73 13,669.40
22 0.005 68.35 13,737.74 386.66 318.31 13,351.08
23 0.005 66.76 13,417.84 386.66 319.90 13,031.18
24 0.005 65.16 13,096.34 386.66 321.50 12,709.68
25 0.005 63.55 12,773.22 386.66 323.11 12,386.56
26 0.005 61.93 12,448.50 386.66 324.73 12,061.84
27 0.005 60.31 12,122.15 386.66 326.35 11,735.49
28 0.005 58.68 11,794.16 386.66 327.98 11,407.50
29 0.005 57.04 11,464.54 386.66 329.62 11,077.88
30 0.005 55.39 11,133.27 386.66 331.27 10,746.61
31 0.005 53.73 10,800.34 386.66 332.93 10,413.68
32 0.005 52.07 10,465.75 386.66 334.59 10,079.09
33 0.005 50.40 10,129.49 386.66 336.26 9,742.83
34 0.005 48.71 9,791.54 386.66 337.95 9,404.88
35 0.005 47.02 9,451.91 386.66 339.64 9,065.25
36 0.005 45.33 9,110.57 386.66 341.33 8,723.91
Late
37 0.00833 72.70 8,796.61 402.56 329.86 8,394.05
i=0.833%
38 0.00833 69.95 8,464.00 402.56 332.61 8,061.43
A = $402.56
39 0.00833 67.18 8,128.61 402.56 335.39 7,726.05
40 0.00833 64.38 7,790.43 402.56 338.18 7,387.87
41 0.00833 61.57 7,449.43 402.56 341.00 7,046.87
42 0.00833 58.72 7,105.59 402.56 343.84 6,703.03
43 0.00833 55.86 6,758.89 402.56 346.71 6,356.32
44 0.00833 52.97 6,409.29 402.56 349.59 6,006.73
45 0.00833 50.06 6,056.79 402.56 352.51 5,654.22
46 0.00833 47.12 5,701.34 402.56 355.45 5,298.78
47 0.00833 44.16 5,342.93 402.56 358.41 4,940.37
48 0.00833 41.17 4,981.54 402.56 361.39 4,578.97
49 0.00833 38.16 4,617.13 402.56 364.41 4,214.57
50 0.00833 35.12 4,249.69 402.56 367.44 3,847.13
51 0.00833 32.06 3,879.19 402.56 370.50 3,476.62
52 0.00833 28.97 3,505.59 402.56 373.59 3,103.03
53 0.00833 25.86 3,128.89 402.56 376.71 2,726.32
54 0.00833 22.72 2,749.04 402.56 379.84 2,346.48
55 0.00833 19.55 2,366.03 402.56 383.01 1,963.47
Additional Problems and FE Exam Review Questions
4.68 i/year = (1 + 0.01)12 –1 = 0.1268 (12.68%)
4.69 Answer is (d)
4.70 Calculate quarterly and semiannual rates to determine correct answer
4.73 PP of month < CP of 6 months. Assume no interperiod compounding
4.74 r = 2% per quarter = 4% per 6–mth period
4.75 Answer is (a)
4.76 A = 500,000(A/F,7%,13)
4.78 P1 = 470(P/A,7%,6) – 50(P/G,7%,6)
F4 = 1691.34(F/P,7%,7) + 200
4.80 Answer is (b)