# Engineering Economic Analysis - 9th Edition

Engineering Economic Analysis
9th Edition
Chapter 4
MORE INTEREST FORMULAS
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
Components of Engineering Economic
Analysis
• Calculation of P,A,and F are fundamental.
• Some problems are more complex and
components:
•
•
•
•
Uniform series
Nominal and effective interest rates
Continuous compounding
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
2
Uniform Payment Series
Compound Amount Factor F
• The future value of an
investment based on
periodic, constant
payments and a constant
interest rate.
F= A(F/A,i,n)
Compound Interest Factors
Uniform Payment Series
Interest rate
P
F
A
10.00% The interest rate may be changed.
\$1.00 A' may be changed.
Compound
Amount
Factor
F/A
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
\$1.00 These values will be modified.
\$2.10
\$3.31
\$4.64
\$6.11
\$7.72
\$9.49
\$11.44
\$13.58
\$15.94
\$18.53
\$21.38
\$24.52
\$27.97
\$31.77
\$35.95
\$40.54
\$45.60
\$51.16
\$57.27
\$64.00
\$71.40
\$79.54
\$88.50
\$98.35
\$109.18
\$121.10
\$134.21
\$148.63
\$164.49
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
3
Example 4-1
At 5%/year
Cash
in
0
0
1
\$500
2
Cash
out
Cash flow calculator
Initial deposit
Annual deposit
Years
Interest rate
\$500
3
\$500
4
\$500
5
\$500
\$2763
F=
\$500(F/A, 5%,5)
= \$500(5.526)
= \$2763
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Deposit
0.00
500.00
500.00
500.00
500.00
500.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
\$0.00
\$500.00
5
5.00%
Cash flow diagram
\$1,000.00
Future worth
\$500.00
\$0.00
0
2
4
6
8
10
12
14
16
18
20
(\$500.00)
(\$2,762.82)
\$
Year
(\$1,000.00)
(\$1,500.00)
(\$2,000.00)
(\$2,500.00)
(\$3,000.00)
Years
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
4
Uniform Payment Series
Sinking Fund Factor
Compound Interest Factors
Uniform Payment Series
Interest rate
• The constant periodic
amount, at a constant
interest rate, that must
be deposited to
accumulate a future
value.
A = F(A/F,i,n)
P
F
A
\$1.00
\$1.00
Sinking
Fund
Factor
A/F
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
8.00%
\$1.00000
\$0.48077
\$0.30803
\$0.22192
\$0.17046
\$0.13632
\$0.11207
\$0.09401
\$0.08008
\$0.06903
\$0.06008
\$0.05270
\$0.04652
\$0.04130
\$0.03683
\$0.03298
\$0.02963
\$0.02670
\$0.02413
\$0.02185
\$0.01983
\$0.01803
\$0.01642
\$0.01498
\$0.01368
\$0.01251
\$0.01145
\$0.01049
\$0.00962
\$0.00883
Compound
Amount
Factor
F/A
\$1.000
\$2.080
\$3.246
\$4.506
\$5.867
\$7.336
\$8.923
\$10.637
\$12.488
\$14.487
\$16.645
\$18.977
\$21.495
\$24.215
\$27.152
\$30.324
\$33.750
\$37.450
\$41.446
\$45.762
\$50.423
\$55.457
\$60.893
\$66.765
\$73.106
\$79.954
\$87.351
\$95.339
\$103.966
\$113.283
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
5
Uniform Payment Series
Capital Recovery Factor
• The series of
uniform payments
that will recover an
initial investment.
A = P(A/P,i,n)
Compound Interest Factors
Uniform Payment Series
P
F
A
Interest rate =
\$345.62
\$1.00
Sinking Fund
Factor
A/F
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10.00%
\$1.00
\$1.00000
\$0.47619
\$0.30211
\$0.21547
\$0.16380
\$0.12961
\$0.10541
\$0.08744
\$0.07364
\$0.06275
\$0.05396
\$0.04676
\$0.04078
\$0.03575
\$0.03147
\$0.02782
\$0.02466
\$0.02193
\$0.01955
\$0.01746
\$0.01562
\$0.01401
\$0.01257
\$0.01130
\$0.01017
\$0.00916
\$0.00826
\$0.00745
\$0.00673
\$0.00608
Capital Recovery
Factor
A/P
\$380.1820
\$199.1430
\$138.9789
\$109.0330
\$91.1737
\$79.3569
\$70.9922
\$64.7844
\$60.0136
\$56.2481
\$53.2127
\$50.7243
\$48.6559
\$46.9166
\$45.4400
\$44.1760
\$43.0864
\$42.1415
\$41.3178
\$40.5964
\$39.9621
\$39.4024
\$38.9071
\$38.4674
\$38.0763
\$37.7275
\$37.4160
\$37.1372
\$36.8874
\$36.6631
Compound Amount
Factor
F/A
\$1.000
\$2.100
\$3.310
\$4.641
\$6.105
\$7.716
\$9.487
\$11.436
\$13.579
\$15.937
\$18.531
\$21.384
\$24.523
\$27.975
\$31.772
\$35.950
\$40.545
\$45.599
\$51.159
\$57.275
\$64.002
\$71.403
\$79.543
\$88.497
\$98.347
\$109.182
\$121.100
\$134.210
\$148.631
\$164.494
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
6
Uniform Payment Series
Present Worth Factor
Compound Interest Factors
Uniform Payment Series
The present value
of a series of
uniform future
payments.
P = A(P/A,i,n)
P
F
A
Interest rate =
\$1.00
\$1.00
\$1.00
Sinking
Fund
Factor
A/F
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10.00%
\$1.00000
\$0.47619
\$0.30211
\$0.21547
\$0.16380
\$0.12961
\$0.10541
\$0.08744
\$0.07364
\$0.06275
\$0.05396
\$0.04676
\$0.04078
\$0.03575
\$0.03147
\$0.02782
\$0.02466
\$0.02193
\$0.01955
\$0.01746
\$0.01562
\$0.01401
\$0.01257
\$0.01130
\$0.01017
\$0.00916
\$0.00826
\$0.00745
\$0.00673
\$0.00608
capital
Recovery
Factor
A/P
\$1.1000
\$0.5762
\$0.4021
\$0.3155
\$0.2638
\$0.2296
\$0.2054
\$0.1874
\$0.1736
\$0.1627
\$0.1540
\$0.1468
\$0.1408
\$0.1357
\$0.1315
\$0.1278
\$0.1247
\$0.1219
\$0.1195
\$0.1175
\$0.1156
\$0.1140
\$0.1126
\$0.1113
\$0.1102
\$0.1092
\$0.1083
\$0.1075
\$0.1067
\$0.1061
Compound
Amount
Factor
F/A
\$1.000
\$2.100
\$3.310
\$4.641
\$6.105
\$7.716
\$9.487
\$11.436
\$13.579
\$15.937
\$18.531
\$21.384
\$24.523
\$27.975
\$31.772
\$35.950
\$40.545
\$45.599
\$51.159
\$57.275
\$64.002
\$71.403
\$79.543
\$88.497
\$98.347
\$109.182
\$121.100
\$134.210
\$148.631
\$164.494
\$1.00
Present
Worth
Factor
P/A
\$0.909
\$1.736
\$2.487
\$3.170
\$3.791
\$4.355
\$4.868
\$5.335
\$5.759
\$6.145
\$6.495
\$6.814
\$7.103
\$7.367
\$7.606
\$7.824
\$8.022
\$8.201
\$8.365
\$8.514
\$8.649
\$8.772
\$8.883
\$8.985
\$9.077
\$9.161
\$9.237
\$9.307
\$9.370
\$9.427
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
7
Example 4-5
Uniform Payment Series
A = \$140/month
i= 1%/month
n= 30 months
• Is the above
equivalent to
\$6800 now?
P
F
A
Interest rate =
\$1.00
\$1.00
\$1.00
Sinking Fund
Factor
A/F
Year
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
60
1.00%
\$1.00000
\$0.49751
\$0.33002
\$0.24628
\$0.19604
\$0.16255
\$0.13863
\$0.12069
\$0.10674
\$0.09558
\$0.08645
\$0.07885
\$0.07241
\$0.06690
\$0.06212
\$0.05794
\$0.05426
\$0.05098
\$0.04805
\$0.04542
\$0.04303
\$0.04086
\$0.03889
\$0.03707
\$0.03541
\$0.03387
\$0.03245
\$0.03112
\$0.02990
\$0.02875
\$0.01224
Capital Recovery
Factor
A/P
\$1.0100
\$0.5075
\$0.3400
\$0.2563
\$0.2060
\$0.1725
\$0.1486
\$0.1307
\$0.1167
\$0.1056
\$0.0965
\$0.0888
\$0.0824
\$0.0769
\$0.0721
\$0.0679
\$0.0643
\$0.0610
\$0.0581
\$0.0554
\$0.0530
\$0.0509
\$0.0489
\$0.0471
\$0.0454
\$0.0439
\$0.0424
\$0.0411
\$0.0399
\$0.0387
\$0.0222
Compound Amount
Factor
F/A
\$1.000
\$2.010
\$3.030
\$4.060
\$5.101
\$6.152
\$7.214
\$8.286
\$9.369
\$10.462
\$11.567
\$12.683
\$13.809
\$14.947
\$16.097
\$17.258
\$18.430
\$19.615
\$20.811
\$22.019
\$23.239
\$24.472
\$25.716
\$26.973
\$28.243
\$29.526
\$30.821
\$32.129
\$33.450
\$34.785
\$81.670
\$140.00
Present Worth
Factor
P/A
\$138.614
\$275.855
\$411.738
\$546.275
\$679.480
\$811.367
\$941.947
\$1,071.235
\$1,199.242
\$1,325.983
\$1,451.468
\$1,575.711
\$1,698.724
\$1,820.518
\$1,941.107
\$2,060.502
\$2,178.715
\$2,295.758
\$2,411.641
\$2,526.377
\$2,639.978
\$2,752.453
\$2,863.815
\$2,974.074
\$3,083.242
\$3,191.329
\$3,298.345
\$3,404.302
\$3,509.210
\$3,613.079
\$6,293.705
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
8
Example 4-6
Cash flow
1
\$100
2
\$100
3
\$100
4
\$0
5
F
Cash flow calculator
Initial deposit
Annual deposit
Start deposits in year
End deposits in year
Years to withdrawal
Interest rate
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Deposit
0.00
100.00
100.00
100.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
\$0.00
\$100.00
1
3
5
15.00%
Cash flow diagram
\$200.00
\$100.00
Future worth
\$0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(\$100.00)
(\$459.24)
\$
F1 = \$100(F/A,15%,3)
= \$347.25
F2 = \$347.25(F/P,15%,2)
= \$459.24
Year
(\$200.00)
(\$300.00)
(\$400.00)
(\$500.00)
Years
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
9
Example 4-7
P = \$20(P/F,15%,2) +
\$30(P/F,15%,2) +
\$20(P/F,15%,2)
= \$46.28
Year
Cash flow
0
P
1
0
2
\$ 20
3
\$ 30
4
\$ 20
Cash flow calculator
Cash flow diagram
Interest rate
15.00%
\$40.00
Deposit
required
\$
(46.28)
\$30.00
Withdrawals
\$
\$
\$
\$
20.00
30.00
20.00
\$20.00
\$10.00
\$0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
\$
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(\$10.00)
(\$20.00)
(\$30.00)
(\$40.00)
(\$50.00)
(\$60.00)
Years
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
0
• A uniform increasing
amount.
• The first cash flow is
always equal to zero.
• G = the difference
between each cash
amount.
G = \$10
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
1
Present Worth Factor
Example 4-8
The equivalent
present value of a
uniformly increasing
amount.
PG = G(P/G,i,n)
Interest
rate =
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
PW =
5.00%
Cash
series
\$120.00
\$150.00
\$180.00
\$210.00
\$240.00
\$766.64
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
2
Uniform Series Factor
Example 4-9
The equivalent
present value of a
uniformly increasing
amount.
AG = G(A/G,i,n)
Interest
rate =
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
AW
6.00%
Cash series
\$100.00
\$200.00
\$300.00
\$400.00
\$500.00
(\$288.36)
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
3
Geometric Series
Present Worth Factor
Example 4-12
The equivalent
present value of a
geometrically
increasing amount.
P = A(P/A,g,i,n)
Interest rate
8.00%
Initial cash
flow
\$100.00
Uniform
rate of cash
flow change
10.00%
Years
5.0
Year
0
1
2
3
4
5
6
7
8
9
10
11
12
PW
Cash series
\$100.00
\$110.00
\$121.00
\$133.10
\$146.41
\$480.43
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
4
Nominal and Effective Interest
• Nominal interest rate/year: the annual interest rate
w/o considering the effect of any compounding.
• 12%/year
• Interest rate/period: the nominal interest rate/year
divided by the number of interest compounding
periods.
• 12%/year/12 months/year = 1%/period
• Effective interest rate/year: the annual interest rate
taking into account the effect of the compounding
periods in the year.
• 12%/year compounded monthly is equivalent to 12.68%/year
compounded yearly
Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc.
1
5
Nominal vs. Effective Rates
A Comparison - A Deposit
Nominal and Effective Interest Rate Comparison - A deposit
Amount deposited
Nominal Interest Rate/Year
\$10,000.00
30.00 %
Yearly
Bi-annually
Quarterly
Monthly
1
2
4
12
Period rate
30.00%
15.00%
7.50%
2.50%
Effective annual rate
30.0000%
32.2500%
33.5469%
34.4889%
Year
Account value
Interest paid
Account value
Interest paid
0
\$10,000.00
\$3,000.00
\$10,000.00
\$3,448.89
1
\$13,000.00
\$3,900.00
\$13,448.89
\$4,638.37
2
\$16,900.00
\$5,070.00
\$18,087.26
\$6,238.09
3
\$21,970.00
\$6,591.00
\$24,325.35
\$8,389.54
4
\$28,561.00
\$8,568.30
\$32,714.90
\$11,283.00
5
\$37,129.30
\$11,138.79
\$43,997.90
\$15,174.38
6
\$48,268.09
\$14,480.43
\$59,172.28
\$20,407.86
7
\$62,748.52
\$18,824.56
\$79,580.14
\$27,446.30
8
\$81,573.07
\$24,471.92
\$107,026.44
\$36,912.22
9
\$106,044.99
\$31,813.50
\$143,938.66
\$49,642.84
10
\$137,858.49
\$41,357.55
\$193,581.50
\$66,764.10
11
\$179,216.04
\$53,764.81
\$260,345.59
\$89,790.29
12
\$232,980.85
\$69,894.26
\$350,135.88
\$120,757.95
13
\$302,875.11
\$90,862.53
\$470,893.83
\$162,406.02
14
\$393,737.64
\$118,121.29
\$633,299.85
\$218,418.04
15
\$511,858.93
\$153,557.68
\$851,717.89
\$293,747.98
16
\$665,416.61
\$199,624.98
\$1,145,465.87
\$395,058.38
17
\$865,041.59
\$259,512.48
\$1,540,524.25
\$531,309.60
18
\$1,124,554.07
\$2,071,833.85
\$714,552.34 1
Engineering Economic
Analysis
- Ninth Edition\$337,366.22
University Press, Inc.
19
\$1,461,920.29
\$438,576.09
\$2,786,386.19
\$960,993.46 6
20
\$1,900,496.38
\$570,148.91
\$3,747,379.65
\$1,292,429.36
Bi-monthly
24
1.25%
34.7351%
Nominal vs. Effective Rates
A Comparison - A Loan Repaid Monthly
Nominal and Effective Interest Rates A Comparison
A loan paid monthly
Amount Borrowed
Nominal Interest Rate/Year
Yearly
Engineering
1
Period rate
12.00%
Effective annual rate
12.00%
Year
Payments
Total repaid
1 (\$11,200.00)
(\$11,200.00)
2
(\$5,916.98)
(\$11,833.96)
3
(\$4,163.49)
(\$12,490.47)
4
(\$3,292.34)
(\$13,169.38)
5
(\$2,774.10)
(\$13,870.49)
6
(\$2,432.26)
(\$14,593.54)
7
(\$2,191.18)
(\$15,338.24)
8
(\$2,013.03)
(\$16,104.23)
9
(\$1,876.79)
(\$16,891.10)
10
(\$1,769.84)
(\$17,698.42)
11
(\$1,684.15)
(\$18,525.69)
12
(\$1,614.37)
(\$19,372.42)
13
(\$1,556.77)
(\$20,238.04)
14
(\$1,508.71)
(\$21,121.97)
15
(\$1,468.24)
(\$22,023.64)
16
(\$1,433.90)
(\$22,942.40)
17
(\$1,404.57)
(\$23,877.64)
18
(\$1,379.37)
(\$24,828.72)
19
(\$1,357.63)
(\$25,794.97)
20
(\$1,338.79)
(\$26,775.76)
21
(\$1,322.40)
(\$27,770.42)
22
(\$1,308.11)
(\$28,778.31)
Economic Analysis - Ninth
Newnan/Eschenbach/Lavelle
23Edition
(\$1,295.60)
(\$29,798.79)
24
(\$1,284.63)
(\$30,831.23)
25
(\$1,275.00)
(\$31,874.99)
26
(\$1,266.52)
(\$32,929.48)
\$10,000.00
12.00 %
Monthly
12
1.00%
12.68%
Payments
Total repaid
(\$888.49)
(\$10,661.85)
(\$470.73)
(\$11,297.63)
(\$332.14)
(\$11,957.15)
(\$263.34)
(\$12,640.24)
(\$222.44)
(\$13,346.67)
(\$195.50)
(\$14,076.14)
(\$176.53)
(\$14,828.30)
(\$162.53)
(\$15,602.73)
(\$151.84)
(\$16,398.97)
(\$143.47)
(\$17,216.51)
(\$136.78)
(\$18,054.80)
(\$131.34)
(\$18,913.24)
(\$126.87)
(\$19,791.19)
(\$123.14)
(\$20,688.02)
(\$120.02)
(\$21,603.03)
(\$117.37)
(\$22,535.52)
(\$115.12)
(\$23,484.80)
(\$113.20)
(\$24,450.13)
(\$111.54)
(\$25,430.79)
(\$110.11)
(\$26,426.07)
(\$108.87)
(\$27,435.23)
(\$107.79)
(\$28,457.57)
2004 by Oxford
University
(\$29,492.39)
(\$106.04)
(\$30,539.00)
(\$105.32)
(\$31,596.72)
(\$104.70)
(\$32,664.92)
Press, Inc.
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