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
require an understanding of added
components:
•
•
•
•
Uniform series
Arithmetic or geometric gradients
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
Arithmetic Gradient
• 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
Arithmetic Gradient
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
Arithmetic Gradient
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)
Geometric Gradient
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
Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford
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)
Copyright($106.86)
2004 by Oxford
University
($29,492.39)
($106.04)
($30,539.00)
($105.32)
($31,596.72)
($104.70)
($32,664.92)
Press, Inc.
1
7