P24 MT2 240813 143329

Toronto Metropolitan University
DEPARTMENT OF ECONOMICS
ECN 801 Principles of Engineering Economics
TIME: 2 HOURS
DATE: Spring/Summer 2024
1. MARR = 6%. You purchase an annual coupon bond for $8418.17. The bond’s face value is
$10,000 and the coupon rate is 7%. You receive your first annual coupon payment immediately
aGer you purchase the bond, and there are 15 years unIl the bond matures. Find the IRR of the
bond (to the closest integer).
a) 6%
b) 7%
c) 8%
d) 9%
e) 10%
f) None of the above
Solution: P = 0 = -8418.17 + 700 + 700(P/A,i*,15) + 10,000(P/F,i*,15)
i* = 10%
2. MARR = 6%. There are two 10-year projects under consideraIon that each provide a required
service. Project A has a first cost of $7044. There are no maintenance costs for the first 2 years.
AGer two years, the maintenance costs are $2000 a year for 8 years. Project B has a first cost of
4715 and also has no maintenance costs for the first 2 years. The maintenance costs aGer that
are $2500 each year for 8 years. You need to choose only one of these projects. Find the
incremental IRR (to the closest integer).
A) 6%
B) 7%
C) 8%
D) 9%
E) 10%
F) None of the above
Solu4on: Incremental Project A – B has t=0 cash flow of -7044 – (-4715) = -2329, and cash
flows of +500 from t= 3 to t = 10.
PW(A – B) = -2329 + 500(P/A,i*,10)(P/F,i*,2) = 0 for i* = 9%
3. MARR = 6%. Suppose that you take on a consulIng project that involves an up-front receipt of
$2000 for work to be delivered. You then provide $3000 of labour services in the first year and
$3000 of labour services in the second year. In the final year you receive the balance owing
which is $4266.43. Find the precise ERR of the project (to the closest integer). Note, the precise
ERR is between 6% and 12%.
A) 6%
B) 7%
C) 8%
D) 9%
E) 10%
F) None of the above
Solu4on: 2000(1.06) – 3000 = -880
PW = 0 = -880 – 3000/(1+ie*) + 4266.32/(1+ie*)2
ie* = 8%
4. MARR = 6%. Suppose that you take on a consulIng project that involves an up-front receipt of
$2000 for work to be delivered. You then provide $3000 of labour services in the first year and
$3000 of labour services in the second year. In the final year you receive the balance owing
which is $4266.43. Find the approximated ERR of the project (to the closest integer). Note, the
approximated ERR is between 6% and 12%.
A) 6%
B) 7%
C) 8%
D) 9%
E) 10%
F) None of the above
Solu4on: F = 2000(1.06)3 + 4266.43 = 3000(1+iea*)2 + 3000(1+iea*)
iea*=7%
5. MARR = 6%. Suppose you purchase an industrial card printer for $30,000 and sell it 8 years
aGer for $2000. The O&M in the first year is $1000, increasing by 6% each year. You expect to
print 5000 cards in the first year, increasing by 500 cards each year thereaGer.
Find the levelized cost within 2 cents.
a) 89 cents
b) 91 cents
c) 95 cents
d) 99 cents
e) None of the above
Solu4on: 0 = -30k + 2k(P/F,6%,8) – 1000 (P/A,6%,6%,8)+ PBE*[5k + 500(A/G,6%,8)](P/A,6%,8)
P = $0.89
The following information pertains to questions 6 and 7:
Salvador Industries bought land and built its plant 20 years ago. The depreciation on the
building is calculated using the declining balance method and a depreciation rate of
15%, with a life of 25 years, and a salvage value of $125,000. Land is not depreciated.
The depreciation for the equipment, all of which was purchased at the same time the
plant was constructed, is calculated using straight-line approach with salvage value of
$75,000. Salvador currently has two outstanding loans: one due at the end of this year,
on December 31,2020, and another one for which the next payment is due in four years.
The value of Retained Earnings is $1,559,596. The values of balance sheet entries
should be calculated as correct to the nearest dollar. MARR = 6%.
6. What is the value of Total LiabiliIes and Owner’s Equity? _________________
Answer: BV20 of Building = 224,000(0.85)20 = 8682
BV20 of Equipment: 460k – 20(460k – 75k)/25 = 152,000
Current Assets = 344k + 2860k + 2002k +162k = 5,368,000
LT Assets Assets = 8682 + 152,000 + 525,000 = 685,682
Total Assets = 5,368,000 + 685,682 = 6,053,682
Total L + OE = 6,053,682
7. What is the Current Ratio (correct to two decimal places)? _____________________
Answer: Total Owner’s Equity = Common Shares + Retained Earnings
= 1,880,000 + 1,559,596 = 3,439,596
Total Liabilities = Total Assets – Total OE
= 6,053,682 – 3,439,596 = 2,614,086
Total L-T Liabilities = 1,220,323 + 323,000 =1,543,000
Total Current Liabilities = Total – LT Liabilities = 2,614,086– 1,543,000 = 1,071,086
Current Ratio = Current Assets / Current Liabilities = 5,368,000/1,071,086 = 5.01
The following information pertains to questions 8 through 11:
8. MARR = 6%. You purchase a machine for $40,000. The market value of the machine
depreciates at 20% per year afterward. There are no operating costs in the Mirst year, but
after that the cost in the second year is $1000, doubling each year thereafter. There is also a
one-time repair cost of $12,000 in the 4th year. Find the value of EAC5.
EAC5 = _____________
Solu4on: EAC5 = 40k(A/P,6%,5) – 40k(0.8)5 (A/F,6%,5) + [1k/(1.06)2 +2k/(1.06)3 + [4k +
12k]/(1.06)4 + 8k/(1.06)5](A/P,6%,5) = 12,208.60
9. Find EAC*.
EAC* = ___________
Solu4on:
EAC2 = 40k(A/P,6%,2) – 40k(0.8)2 (A/F,6%,2) + [1k/(1.06)2] (A/P,6%,2) = 9875.78
EAC3 = 40k(A/P,6%,3) – 40k(0.8)3 (A/F,6%,3) + [1k/(1.06)2 +2k/(1.06)3](A/P,6%,3) = 9492.60
EAC4 = 40k(A/P,6%,4) – 40k(0.8)4 (A/F,6%,4) + [1k/(1.06)2 +2k/(1.06)3 + [4k +
12k]/(1.06)4](A/P,6%,4) = 12,197.28
Therefore EAC* = 9492.60
10. MARR = 6%. You purchased a machine 3 years ago for $40,000. The market value of the
machine depreciates at 20% per year aGerward. There were no operaIng costs in the first year,
but aGer that the cost in the second year was $1000, doubling each year thereaGer. There is
also a one-Ime repair cost of $12,000 in the 4th year.
Find the Marginal Cost of keeping the asset for this coming year (its fourth year).
The new asset has a first cost of $20,000. The resale value of the new asset falls by 10% each
year, and annual O&M costs are 10k in the first year, increasing by 20% each year. All the costs
and the resale market value informaIon that you esImated when you first purchased the asset
are sIll valid. Find the marginal cost of keeping the defender for one year.
MC4 = _______________
Solu4on:
MC4 = 40k(0.83)(1.06) + (4k + 12k) – 40k(0.84) = 21,324.8
11. MARR = 6%. You purchased a machine 3 years ago for $40,000. The market value of the
machine depreciates at 20% per year. There were no operaIng costs in the asset's first year, but
aGer that the cost in the second year was $1000, doubling each year thereaGer. There is also a
one-Ime repair cost of $12,000 in the 4th year.
Now that the asset is 3 years old, you are considering replacing the asset with a new machine
that has become available. find the economic life of the defender going forward.
N* = _____
Solu4on:
EAC1 = MC4 = 40k(0.83)(1.06) + (4k + 12k) – 40k(0.84) = 21,324.8
EAC2 = 40k(0.83)(A/P,6%,2) – 40k(0.85)(A/F,6%,2) + (16k/1.06 + 8k/(1.062)](A/P,6%,2) =
16,924.43
EAC3 = 40k(0.83)(A/P,6%,3) – 40k(0.86)(A/F,6%,3) + (16k/1.06 + 8k/(1.062) +
16k/(1.063)](A/P,6%,3) = 17,704.44
Therefore N*= 2 and EAC* = 16,924.43
12. MARR = 6%. A new industrial spindle has recently become available. You are considering this
new asset to replace your current spindle. The new spindle has a first cost of $20,000. Its resale
value is expected to fall by 10% each year, and annual O&M costs are projected to be 10k in the
first year, increasing by 20% each year thereaGer. Find the asset’s equivalent annual cost at its
economic life. (within $100).
A) 12,600
B) 12,800
C) 13,000
D) 13,200
E) None of the above
Solu4on:
Challenger sa4sfies one-year principle:
EAC* = 20k(1.06) – 20k(0.9) + 10k = 13,200 at N*=1
Verify: EAC2 = 20k(A/P,6%,2) – 20K(0.92)(A/F,6%,2) + [10K/1.06 + 12K(1.062)](A/P,6%,2) =
14,015.61 > 13,200
Compound Interest Factors for Discrete Compounding, Discrete Cash Flows
i 5 6%
473
Discrete Compounding, Discrete Cash Flows
Single Payment
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.0600
1.1236
1.1910
1.2625
1.3382
0.94340
0.89000
0.83962
0.79209
0.74726
1.0000
0.48544
0.31411
0.22859
0.17740
1.0000
2.0600
3.1836
4.3746
5.6371
1.0600
0.54544
0.37411
0.28859
0.23740
0.94340
1.8334
2.6730
3.4651
4.2124
0.00000
0.48544
0.96118
1.4272
1.8836
6
7
8
9
10
1.4185
1.5036
1.5938
1.6895
1.7908
0.70496
0.66506
0.62741
0.59190
0.55839
0.14336
0.11914
0.10104
0.08702
0.07587
6.9753
8.3938
9.8975
11.491
13.181
0.20336
0.17914
0.16104
0.14702
0.13587
4.9173
5.5824
6.2098
6.8017
7.3601
2.3304
2.7676
3.1952
3.6133
4.0220
11
12
13
14
15
1.8983
2.0122
2.1329
2.2609
2.3966
0.52679
0.49697
0.46884
0.44230
0.41727
0.06679
0.05928
0.05296
0.04758
0.04296
14.972
16.870
18.882
21.015
23.276
0.12679
0.11928
0.11296
0.10758
0.10296
7.8869
8.3838
8.8527
9.2950
9.7122
4.4213
4.8113
5.1920
5.5635
5.9260
16
17
18
19
20
2.5404
2.6928
2.8543
3.0256
3.2071
0.39365
0.37136
0.35034
0.33051
0.31180
0.03895
0.03544
0.03236
0.02962
0.02718
25.673
28.213
30.906
33.760
36.786
0.09895
0.09544
0.09236
0.08962
0.08718
10.106
10.477
10.828
11.158
11.470
6.2794
6.6240
6.9597
7.2867
7.6051
21
22
23
24
25
3.3996
3.6035
3.8197
4.0489
4.2919
0.29416
0.27751
0.26180
0.24698
0.23300
0.02500
0.02305
0.02128
0.01968
0.01823
39.993
43.392
46.996
50.816
54.865
0.08500
0.08305
0.08128
0.07968
0.07823
11.764
12.042
12.303
12.550
12.783
7.9151
8.2166
8.5099
8.7951
9.0722
26
27
28
29
30
4.5494
4.8223
5.1117
5.4184
5.7435
0.21981
0.20737
0.19563
0.18456
0.17411
0.01690
0.01570
0.01459
0.01358
0.01265
59.156
63.706
68.528
73.640
79.058
0.07690
0.07570
0.07459
0.07358
0.07265
13.003
13.211
13.406
13.591
13.765
9.3414
9.6029
9.8568
10.103
10.342
31
32
33
34
35
6.0881
6.4534
6.8406
7.2510
7.6861
0.16425
0.15496
0.14619
0.13791
0.13011
0.01179
0.01100
0.01027
0.00960
0.00897
84.802
90.890
97.343
104.18
111.43
0.07179
0.07100
0.07027
0.06960
0.06897
13.929
14.084
14.230
14.368
14.498
10.574
10.799
11.017
11.228
11.432
40
45
50
55
10.286
13.765
18.420
24.650
0.09722
0.07265
0.05429
0.04057
0.00646
0.00470
0.00344
0.00254
154.76
212.74
290.34
394.17
0.06646
0.06470
0.06344
0.06254
15.046
15.456
15.762
15.991
12.359
13.141
13.796
14.341
60
65
70
75
32.988
44.145
59.076
79.057
0.03031
0.02265
0.01693
0.01265
0.00188
0.00139
0.00103
0.00077
533.13
719.08
967.93
1300.9
0.06188
0.06139
0.06103
0.06077
16.161
16.289
16.385
16.456
14.791
15.160
15.461
15.706
80
85
90
95
100
105.80
141.58
189.46
253.55
339.30
0.00945
0.00706
0.00528
0.00394
0.00295
0.00057
0.00043
0.00032
0.00024
0.00018
1746.6
2343.0
3141.1
4209.1
5638.4
0.06057
0.06043
0.06032
0.06024
0.06018
16.509
16.549
16.579
16.601
16.618
15.903
16.062
16.189
16.290
16.371
Z01_FRAS8826_07_SE_APPA.indd 473
28/07/20 3:45 PM
474 Appendix A
i 5 7%
Discrete Compounding, Discrete Cash Flows
Single Payment
Z01_FRAS8826_07_SE_APPA.indd 474
Uniform Series
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1.0700
0.55309
0.38105
0.29523
0.24389
0.93458
1.8080
2.6243
3.3872
4.1002
0.00000
0.48309
0.95493
1.4155
1.8650
7.1533
8.6540
10.260
11.978
13.816
0.20980
0.18555
0.16747
0.15349
0.14238
4.7665
5.3893
5.9713
6.5152
7.0236
2.3032
2.7304
3.1465
3.5517
3.9461
0.06336
0.05590
0.04965
0.04434
0.03979
15.784
17.888
20.141
22.550
25.129
0.13336
0.12590
0.11965
0.11434
0.10979
7.4987
7.9427
8.3577
8.7455
9.1079
4.3296
4.7025
5.0648
5.4167
5.7583
0.33873
0.31657
0.29586
0.27651
0.25842
0.03586
0.03243
0.02941
0.02675
0.02439
27.888
30.840
33.999
37.379
40.995
0.10586
0.10243
0.09941
0.09675
0.09439
9.4466
9.7632
10.059
10.336
10.594
6.0897
6.4110
6.7225
7.0242
7.3163
4.1406
4.4304
4.7405
5.0724
5.4274
0.24151
0.22571
0.21095
0.19715
0.18425
0.02229
0.02041
0.01871
0.01719
0.01581
44.865
49.006
53.436
58.177
63.249
0.09229
0.09041
0.08871
0.08719
0.08581
10.836
11.061
11.272
11.469
11.654
7.5990
7.8725
8.1369
8.3923
8.6391
26
27
28
29
30
5.8074
6.2139
6.6488
7.1143
7.6123
0.17220
0.16093
0.15040
0.14056
0.13137
0.01456
0.01343
0.01239
0.01145
0.01059
68.676
74.484
80.698
87.347
94.461
0.08456
0.08343
0.08239
0.08145
0.08059
11.826
11.987
12.137
12.278
12.409
8.8773
9.1072
9.3289
9.5427
9.7487
31
32
33
34
35
8.1451
8.7153
9.3253
9.9781
10.677
0.12277
0.11474
0.10723
0.10022
0.09366
0.00980
0.00907
0.00841
0.00780
0.00723
102.07
110.22
118.93
128.26
138.24
0.07980
0.07907
0.07841
0.07780
0.07723
12.532
12.647
12.754
12.854
12.948
9.9471
10.138
10.322
10.499
10.669
40
45
50
55
14.974
21.002
29.457
41.315
0.06678
0.04761
0.03395
0.02420
0.00501
0.00350
0.00246
0.00174
199.64
285.75
406.53
575.93
0.07501
0.07350
0.07246
0.07174
13.332
13.606
13.801
13.940
11.423
12.036
12.529
12.921
60
65
70
75
57.946
81.273
113.99
159.88
0.01726
0.01230
0.00877
0.00625
0.00123
0.00087
0.00062
0.00044
813.52
1146.8
1614.1
2269.7
0.07123
0.07087
0.07062
0.07044
14.039
14.110
14.160
14.196
13.232
13.476
13.666
13.814
80
85
90
95
100
224.23
314.50
441.10
618.67
867.72
0.00446
0.00318
0.00227
0.00162
0.00115
0.00031
0.00022
0.00016
0.00011
0.00008
3189.1
4478.6
6287.2
8823.9
12382.0
0.07031
0.07022
0.07016
0.07011
0.07008
14.222
14.240
14.253
14.263
14.269
13.927
14.015
14.081
14.132
14.170
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
1
2
3
4
5
1.0700
1.1449
1.2250
1.3108
1.4026
0.93458
0.87344
0.81630
0.76290
0.71299
1.0000
0.48309
0.31105
0.22523
0.17389
1.0000
2.0700
3.2149
4.4399
5.7507
6
7
8
9
10
1.5007
1.6058
1.7182
1.8385
1.9672
0.66634
0.62275
0.58201
0.54393
0.50835
0.13980
0.11555
0.09747
0.08349
0.07238
11
12
13
14
15
2.1049
2.2522
2.4098
2.5785
2.7590
0.47509
0.44401
0.41496
0.38782
0.36245
16
17
18
19
20
2.9522
3.1588
3.3799
3.6165
3.8697
21
22
23
24
25
28/07/20 3:45 PM
Compound Interest Factors for Discrete Compounding, Discrete Cash Flows
i 5 8%
475
Discrete Compounding, Discrete Cash Flows
Single Payment
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.0800
1.1664
1.2597
1.3605
1.4693
0.92593
0.85734
0.79383
0.73503
0.68058
1.0000
0.48077
0.30803
0.22192
0.17046
1.0000
2.0800
3.2464
4.5061
5.8666
1.0800
0.56077
0.38803
0.30192
0.25046
0.92593
1.7833
2.5771
3.3121
3.9927
0.00000
0.48077
0.94874
1.4040
1.8465
6
7
8
9
10
1.5869
1.7138
1.8509
1.9990
2.1589
0.63017
0.58349
0.54027
0.50025
0.46319
0.13632
0.11207
0.09401
0.08008
0.06903
7.3359
8.9228
10.637
12.488
14.487
0.21632
0.19207
0.17401
0.16008
0.14903
4.6229
5.2064
5.7466
6.2469
6.7101
2.2763
2.6937
3.0985
3.4910
3.8713
11
12
13
14
15
2.3316
2.5182
2.7196
2.9372
3.1722
0.42888
0.39711
0.36770
0.34046
0.31524
0.06008
0.05270
0.04652
0.04130
0.03683
16.645
18.977
21.495
24.215
27.152
0.14008
0.13270
0.12652
0.12130
0.11683
7.1390
7.5361
7.9038
8.2442
8.5595
4.2395
4.5957
4.9402
5.2731
5.5945
16
17
18
19
20
3.4259
3.7000
3.9960
4.3157
4.6610
0.29189
0.27027
0.25025
0.23171
0.21455
0.03298
0.02963
0.02670
0.02413
0.02185
30.324
33.750
37.450
41.446
45.762
0.11298
0.10963
0.10670
0.10413
0.10185
8.8514
9.1216
9.3719
9.6036
9.8181
5.9046
6.2037
6.4920
6.7697
7.0369
21
22
23
24
25
5.0338
5.4365
5.8715
6.3412
6.8485
0.19866
0.18394
0.17032
0.15770
0.14602
0.01983
0.01803
0.01642
0.01498
0.01368
50.423
55.457
60.893
66.765
73.106
0.09983
0.09803
0.09642
0.09498
0.09368
10.017
10.201
10.371
10.529
10.675
7.2940
7.5412
7.7786
8.0066
8.2254
26
27
28
29
30
7.3964
7.9881
8.6271
9.3173
10.063
0.13520
0.12519
0.11591
0.10733
0.09938
0.01251
0.01145
0.01049
0.00962
0.00883
79.954
87.351
95.339
103.97
113.28
0.09251
0.09145
0.09049
0.08962
0.08883
10.810
10.935
11.051
11.158
11.258
8.4352
8.6363
8.8289
9.0133
9.1897
31
32
33
34
35
10.868
11.737
12.676
13.690
14.785
0.09202
0.08520
0.07889
0.07305
0.06763
0.00811
0.00745
0.00685
0.00630
0.00580
123.35
134.21
145.95
158.63
172.32
0.08811
0.08745
0.08685
0.08630
0.08580
11.350
11.435
11.514
11.587
11.655
9.3584
9.5197
9.6737
9.8208
9.9611
40
45
50
55
21.725
31.920
46.902
68.914
0.04603
0.03133
0.02132
0.01451
0.00386
0.00259
0.00174
0.00118
259.06
386.51
573.77
848.92
0.08386
0.08259
0.08174
0.08118
11.925
12.108
12.233
12.319
10.570
11.045
11.411
11.690
60
65
70
75
101.26
148.78
218.61
321.20
0.00988
0.00672
0.00457
0.00311
0.00080
0.00054
0.00037
0.00025
1253.2
1847.2
2720.1
4002.6
0.08080
0.08054
0.08037
0.08025
12.377
12.416
12.443
12.461
11.902
12.060
12.178
12.266
80
85
90
95
100
471.95
693.46
1018.9
1497.1
2199.8
0.00212
0.00144
0.00098
0.00067
0.00045
0.00017
0.00012
0.00008
0.00005
0.00004
5886.9
8655.7
12724.0
18702.0
27485.0
0.08017
0.08012
0.08008
0.08005
0.08004
12.474
12.482
12.488
12.492
12.494
12.330
12.377
12.412
12.437
12.455
Z01_FRAS8826_07_SE_APPA.indd 475
28/07/20 3:45 PM
476 Appendix A
i 5 9%
Discrete Compounding, Discrete Cash Flows
Single Payment
Z01_FRAS8826_07_SE_APPA.indd 476
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.0900
1.1881
1.2950
1.4116
1.5386
0.91743
0.84168
0.77218
0.70843
0.64993
1.0000
0.47847
0.30505
0.21867
0.16709
1.0000
2.0900
3.2781
4.5731
5.9847
1.0900
0.56847
0.39505
0.30867
0.25709
0.91743
1.7591
2.5313
3.2397
3.8897
0.00000
0.47847
0.94262
1.3925
1.8282
6
7
8
9
10
1.6771
1.8280
1.9926
2.1719
2.3674
0.59627
0.54703
0.50187
0.46043
0.42241
0.13292
0.10869
0.09067
0.07680
0.06582
7.5233
9.2004
11.028
13.021
15.193
0.22292
0.19869
0.18067
0.16680
0.15582
4.4859
5.0330
5.5348
5.9952
6.4177
2.2498
2.6574
3.0512
3.4312
3.7978
11
12
13
14
15
2.5804
2.8127
3.0658
3.3417
3.6425
0.38753
0.35553
0.32618
0.29925
0.27454
0.05695
0.04965
0.04357
0.03843
0.03406
17.560
20.141
22.953
26.019
29.361
0.14695
0.13965
0.13357
0.12843
0.12406
6.8052
7.1607
7.4869
7.7862
8.0607
4.1510
4.4910
4.8182
5.1326
5.4346
16
17
18
19
20
3.9703
4.3276
4.7171
5.1417
5.6044
0.25187
0.23107
0.21199
0.19449
0.17843
0.03030
0.02705
0.02421
0.02173
0.01955
33.003
36.974
41.301
46.018
51.160
0.12030
0.11705
0.11421
0.11173
0.10955
8.3126
8.5436
8.7556
8.9501
9.1285
5.7245
6.0024
6.2687
6.5236
6.7674
21
22
23
24
25
6.1088
6.6586
7.2579
7.9111
8.6231
0.16370
0.15018
0.13778
0.12640
0.11597
0.01762
0.01590
0.01438
0.01302
0.01181
56.765
62.873
69.532
76.790
84.701
0.10762
0.10590
0.10438
0.10302
0.10181
9.2922
9.4424
9.5802
9.7066
9.8226
7.0006
7.2232
7.4357
7.6384
7.8316
26
27
28
29
30
9.3992
10.245
11.167
12.172
13.268
0.10639
0.09761
0.08955
0.08215
0.07537
0.01072
0.00973
0.00885
0.00806
0.00734
93.324
102.72
112.97
124.14
136.31
0.10072
0.09973
0.09885
0.09806
0.09734
9.9290
10.027
10.116
10.198
10.274
8.0156
8.1906
8.3571
8.5154
8.6657
31
32
33
34
35
14.462
15.763
17.182
18.728
20.414
0.06915
0.06344
0.05820
0.05339
0.04899
0.00669
0.00610
0.00556
0.00508
0.00464
149.58
164.04
179.80
196.98
215.71
0.09669
0.09610
0.09556
0.09508
0.09464
10.343
10.406
10.464
10.518
10.567
8.8083
8.9436
9.0718
9.1933
9.3083
40
45
50
55
31.409
48.327
74.358
114.41
0.03184
0.02069
0.01345
0.00874
0.00296
0.00190
0.00123
0.00079
337.88
525.86
815.08
1260.1
0.09296
0.09190
0.09123
0.09079
10.757
10.881
10.962
11.014
9.7957
10.160
10.430
10.626
60
65
70
75
176.03
270.85
416.73
641.19
0.00568
0.00369
0.00240
0.00156
0.00051
0.00033
0.00022
0.00014
1944.8
2998.3
4619.2
7113.2
0.09051
0.09033
0.09022
0.09014
11.048
11.070
11.084
11.094
10.768
10.870
10.943
10.994
80
85
90
95
100
986.55
1517.9
2335.5
3593.5
5529.0
0.00101
0.00066
0.00043
0.00028
0.00018
0.00009
0.00006
0.00004
0.00003
0.00002
10951.0
16855.0
25939.0
39917.0
61423.0
0.09009
0.09006
0.09004
0.09003
0.09002
11.100
11.104
11.106
11.108
11.109
11.030
11.055
11.073
11.085
11.093
28/07/20 3:45 PM
Compound Interest Factors for Discrete Compounding, Discrete Cash Flows
i 5 10%
477
Discrete Compounding, Discrete Cash Flows
Single Payment
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.1000
1.2100
1.3310
1.4641
1.6105
0.90909
0.82645
0.75131
0.68301
0.62092
1.0000
0.47619
0.30211
0.21547
0.16380
1.0000
2.1000
3.3100
4.6410
6.1051
1.1000
0.57619
0.40211
0.31547
0.26380
0.90909
1.7355
2.4869
3.1699
3.7908
0.00000
0.47619
0.93656
1.3812
1.8101
6
7
8
9
10
1.7716
1.9487
2.1436
2.3579
2.5937
0.56447
0.51316
0.46651
0.42410
0.38554
0.12961
0.10541
0.08744
0.07364
0.06275
7.7156
9.4872
11.436
13.579
15.937
0.22961
0.20541
0.18744
0.17364
0.16275
4.3553
4.8684
5.3349
5.7590
6.1446
2.2236
2.6216
3.0045
3.3724
3.7255
11
12
13
14
15
2.8531
3.1384
3.4523
3.7975
4.1772
0.35049
0.31863
0.28966
0.26333
0.23939
0.05396
0.04676
0.04078
0.03575
0.03147
18.531
21.384
24.523
27.975
31.772
0.15396
0.14676
0.14078
0.13575
0.13147
6.4951
6.8137
7.1034
7.3667
7.6061
4.0641
4.3884
4.6988
4.9955
5.2789
16
17
18
19
20
4.5950
5.0545
5.5599
6.1159
6.7275
0.21763
0.19784
0.17986
0.16351
0.14864
0.02782
0.02466
0.02193
0.01955
0.01746
35.950
40.545
45.599
51.159
57.275
0.12782
0.12466
0.12193
0.11955
0.11746
7.8237
8.0216
8.2014
8.3649
8.5136
5.5493
5.8071
6.0526
6.2861
6.5081
21
22
23
24
25
7.4002
8.1403
8.9543
9.8497
10.835
0.13513
0.12285
0.11168
0.10153
0.09230
0.01562
0.01401
0.01257
0.01130
0.01017
64.002
71.403
79.543
88.497
98.347
0.11562
0.11401
0.11257
0.11130
0.11017
8.6487
8.7715
8.8832
8.9847
9.0770
6.7189
6.9189
7.1085
7.2881
7.4580
26
27
28
29
30
11.918
13.110
14.421
15.863
17.449
0.08391
0.07628
0.06934
0.06304
0.05731
0.00916
0.00826
0.00745
0.00673
0.00608
109.18
121.10
134.21
148.63
164.49
0.10916
0.10826
0.10745
0.10673
0.10608
9.1609
9.2372
9.3066
9.3696
9.4269
7.6186
7.7704
7.9137
8.0489
8.1762
31
32
33
34
35
19.194
21.114
23.225
25.548
28.102
0.05210
0.04736
0.04306
0.03914
0.03558
0.00550
0.00497
0.00450
0.00407
0.00369
181.94
201.14
222.25
245.48
271.02
0.10550
0.10497
0.10450
0.10407
0.10369
9.4790
9.5264
9.5694
9.6086
9.6442
8.2962
8.4091
8.5152
8.6149
8.7086
40
45
50
55
45.259
72.890
117.39
189.06
0.02209
0.01372
0.00852
0.00529
0.00226
0.00139
0.00086
0.00053
442.59
718.90
1163.9
1880.6
0.10226
0.10139
0.10086
0.10053
9.7791
9.8628
9.9148
9.9471
9.0962
9.3740
9.5704
9.7075
60
65
70
75
304.48
490.37
789.75
1271.9
0.00328
0.00204
0.00127
0.00079
0.00033
0.00020
0.00013
0.00008
3034.8
4893.7
7887.5
12709.0
0.10033
0.10020
0.10013
0.10008
9.9672
9.9796
9.9873
9.9921
9.8023
9.8672
9.9113
9.9410
Z01_FRAS8826_07_SE_APPA.indd 477
28/07/20 3:45 PM
478 Appendix A
i 5 11%
Discrete Compounding, Discrete Cash Flows
Single Payment
Z01_FRAS8826_07_SE_APPA.indd 478
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.1100
1.2321
1.3676
1.5181
1.6851
0.90090
0.81162
0.73119
0.65873
0.59345
1.0000
0.47393
0.29921
0.21233
0.16057
1.0000
2.1100
3.3421
4.7097
6.2278
1.1100
0.58393
0.40921
0.32233
0.27057
0.90090
1.7125
2.4437
3.1024
3.6959
0.00000
0.47393
0.93055
1.3700
1.7923
6
7
8
9
10
1.8704
2.0762
2.3045
2.5580
2.8394
0.53464
0.48166
0.43393
0.39092
0.35218
0.12638
0.10222
0.08432
0.07060
0.05980
7.9129
9.783
11.859
14.164
16.722
0.23638
0.21222
0.19432
0.18060
0.16980
4.2305
4.7122
5.1461
5.5370
5.8892
2.1976
2.5863
2.9585
3.3144
3.6544
11
12
13
14
15
3.1518
3.4985
3.8833
4.3104
4.7846
0.31728
0.28584
0.25751
0.23199
0.20900
0.05112
0.04403
0.03815
0.03323
0.02907
19.561
22.713
26.212
30.095
34.405
0.16112
0.15403
0.14815
0.14323
0.13907
6.2065
6.4924
6.7499
6.9819
7.1909
3.9788
4.2879
4.5822
4.8619
5.1275
16
17
18
19
20
5.3109
5.8951
6.5436
7.2633
8.0623
0.18829
0.16963
0.15282
0.13768
0.12403
0.02552
0.02247
0.01984
0.01756
0.01558
39.190
44.501
50.396
56.939
64.203
0.13552
0.13247
0.12984
0.12756
0.12558
7.3792
7.5488
7.7016
7.8393
7.9633
5.3794
5.6180
5.8439
6.0574
6.2590
21
22
23
24
25
8.949
9.934
11.026
12.239
13.585
0.11174
0.10067
0.09069
0.08170
0.07361
0.01384
0.01231
0.01097
0.00979
0.00874
72.265
81.214
91.15
102.17
114.41
0.12384
0.12231
0.12097
0.11979
0.11874
8.0751
8.1757
8.2664
8.3481
8.4217
6.4491
6.6283
6.7969
6.9555
7.1045
26
27
28
29
30
15.080
16.739
18.580
20.624
22.892
0.06631
0.05974
0.05382
0.04849
0.04368
0.00781
0.00699
0.00626
0.00561
0.00502
128.00
143.08
159.82
178.40
199.02
0.11781
0.11699
0.11626
0.11561
0.11502
8.4881
8.5478
8.6016
8.6501
8.6938
7.2443
7.3754
7.4982
7.6131
7.7206
31
32
33
34
35
25.410
28.206
31.308
34.752
38.575
0.03935
0.03545
0.03194
0.02878
0.02592
0.00451
0.00404
0.00363
0.00326
0.00293
221.91
247.32
275.53
306.84
341.59
0.11451
0.11404
0.11363
0.11326
0.11293
8.7331
8.7686
8.8005
8.8293
8.8552
7.8210
7.9147
8.0021
8.0836
8.1594
40
45
50
55
65.001
109.53
184.56
311.00
0.01538
0.00913
0.00542
0.00322
0.00172
0.00101
0.00060
0.00035
581.83
986.6
1668.8
2818.2
0.11172
0.11101
0.11060
0.11035
8.9511
9.0079
9.0417
9.0617
8.4659
8.6763
8.8185
8.9135
28/07/20 3:45 PM
Compound Interest Factors for Discrete Compounding, Discrete Cash Flows
i 5 12%
479
Discrete Compounding, Discrete Cash Flows
Single Payment
Uniform Series
Compound
Amount
Factor
Present
Worth
Factor
Sinking
Fund
Factor
Uniform
Series
Factor
Capital
Recovery
Factor
Series
Present
Worth Factor
Arithmetic
Gradient
Series
Factor
N
(F/P,i,N)
(P/F,i,N)
(A/F,i,N)
(F/A,i,N)
(A/P,i,N)
(P/A,i,N)
(A/G,i,N)
1
2
3
4
5
1.1200
1.2544
1.4049
1.5735
1.7623
0.89286
0.79719
0.71178
0.63552
0.56743
1.0000
0.47170
0.29635
0.20923
0.15741
1.0000
2.1200
3.3744
4.7793
6.3528
1.1200
0.59170
0.41635
0.32923
0.27741
0.89286
1.6901
2.4018
3.0373
3.6048
0.00000
0.47170
0.92461
1.3589
1.7746
6
7
8
9
10
1.9738
2.2107
2.4760
2.7731
3.1058
0.50663
0.45235
0.40388
0.36061
0.32197
0.12323
0.09912
0.08130
0.06768
0.05698
8.1152
10.089
12.300
14.776
17.549
0.24323
0.21912
0.20130
0.18768
0.17698
4.1114
4.5638
4.9676
5.3282
5.6502
2.1720
2.5515
2.9131
3.2574
3.5847
11
12
13
14
15
3.4785
3.8960
4.3635
4.8871
5.4736
0.28748
0.25668
0.22917
0.20462
0.18270
0.04842
0.04144
0.03568
0.03087
0.02682
20.655
24.133
28.029
32.393
37.280
0.16842
0.16144
0.15568
0.15087
0.14682
5.9377
6.1944
6.4235
6.6282
6.8109
3.8953
4.1897
4.4683
4.7317
4.9803
16
17
18
19
20
6.1304
6.8660
7.6900
8.6128
9.6463
0.16312
0.14564
0.13004
0.11611
0.10367
0.02339
0.02046
0.01794
0.01576
0.01388
42.753
48.884
55.750
63.440
72.052
0.14339
0.14046
0.13794
0.13576
0.13388
6.9740
7.1196
7.2497
7.3658
7.4694
5.2147
5.4353
5.6427
5.8375
6.0202
21
22
23
24
25
10.804
12.100
13.552
15.179
17.000
0.09256
0.08264
0.07379
0.06588
0.05882
0.01224
0.01081
0.00956
0.00846
0.00750
81.699
92.503
104.60
118.16
133.33
0.13224
0.13081
0.12956
0.12846
0.12750
7.5620
7.6446
7.7184
7.7843
7.8431
6.1913
6.3514
6.5010
6.6406
6.7708
26
27
28
29
30
19.040
21.325
23.884
26.750
29.960
0.05252
0.04689
0.04187
0.03738
0.03338
0.00665
0.00590
0.00524
0.00466
0.00414
150.33
169.37
190.70
214.58
241.33
0.12665
0.12590
0.12524
0.12466
0.12414
7.8957
7.9426
7.9844
8.0218
8.0552
6.8921
7.0049
7.1098
7.2071
7.2974
31
32
33
34
35
33.555
37.582
42.092
47.143
52.800
0.02980
0.02661
0.02376
0.02121
0.01894
0.00369
0.00328
0.00292
0.00260
0.00232
271.29
304.85
342.43
384.52
431.66
0.12369
0.12328
0.12292
0.12260
0.12232
8.0850
8.1116
8.1354
8.1566
8.1755
7.3811
7.4586
7.5302
7.5965
7.6577
40
45
50
55
93.051
163.99
289.00
509.32
0.01075
0.00610
0.00346
0.00196
0.00130
0.00074
0.00042
0.00024
767.09
1358.2
2400.0
4236.0
0.12130
0.12074
0.12042
0.12024
8.2438
8.2825
8.3045
8.3170
7.8988
8.0572
8.1597
8.2251
Z01_FRAS8826_07_SE_APPA.indd 479
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Appendix D
List of Formulas
After-tax IRR:
IRRafter@tax ≅ IRRbefore@tax * (1 - t)
After-tax MARR:
MARRafter@tax ≅ MARRbefore@tax * (1 - t)
PW(benefits)
BCR =
PW(costs)
Book Value, Declining-Balance:
BVdb(n) = P(1 - d)n
Book Value, Straight-Line:
Capital Tax Factor:
CTF = 1 -
P - S
b
N
td(1 + i>2)
(i + d )(1 + i)
Capital Salvage Factor:
CSF = 1 -
td
(i + d )
Capitalized Value:
P =
(F>A,i,N) =
A
i
(A>P,i,N) =
A = (P - S)(A>P,i,N) + Si
Compound Interest:
• Series Present Worth Factor
(P>A,i,N) =
(1 + i)N - 1
i(1 + i)N
• Arithmetic Gradient to Annuity
Conversion Factor
(A>G,i,N) =
1
N
i
(1 + i)N - 1
• Geometric Gradient Series to
Present Worth Conversion Factor
(P>A,g,i,N) =
F = P(1 + i)
Compound Interest Factors:
• Compound Amount Factor
(F>P,i,N) = (1 + i)N
• Present Worth Factor
1
(1 + i)N
• Sinking Fund Factor
i
(1 + i)N - 1
(P>A,i°,N)
1 + g
(P>A,g,i,N) = °
(1 + i°)N - 1
i°(1 + i°)N
1 + i
- 1
1 + g
¢
1
1 + g
Depreciation Amount, Straight Line:
Dsl(n) =
N
(A>F,i,N) =
i(1 + i)N
(1 + i)N - 1
i° =
Capital Recovery Formula:
(P>F,i,N) =
(1 + i)N - 1
i
• Capital Recovery Factor
Benefit–Cost Ratio:
BVsl(n) = P - n a
• Uniform Series Compound Amount
Factor
P - S
N
Depreciation Amount, Declining
Balance:
Ddb(n) = BVdb(n - 1) * d
Depreciation Rate:
d = 1 -
n S
AP
Effective Interest Rate:
ie = a1 +
r m
b - 1 or
m
ie = (1 + is)m - 1
494
Z04_FRAS8826_07_SE_APPD.indd 494
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List of Formulas
Effective Interest Rate for Continuous
Compounding:
ie = e r - 1
Expected Value of the Discrete
Random Variable:
E(x) = a xip(xi)
• Acid test ratio =
Quick assets
Current liabilities
• Current ratio =
Current assets
Current liabilities
Total assets
Sales
Inventories
• Return on total assets =
Profits after taxes
Total assets
Growth-Adjusted Interest Rate:
Internal Rate of Return:
a (1 + i*)t = 0 or
(Rt - Dt)
t=0
= a
Z04_FRAS8826_07_SE_APPD.indd 495
y* - y1
y2 - y1
Modified Benefit–Cost Ratio:
BCRM =
d
PW(benefits) - PW(operating costs)
PW(capital costs )
Payback period =
T
Dt
t
t = 0 (1 + i*)
First cost
Annual savings
Real Dollars:
R0,N =
RN =
AN
I0,N >100
AN
(1 + f )N
RN = AN(P>F,f,N)
Real MARR:
MARRR =
1 + MARRC
- 1
1 + f
Real Interest Rate:
i′ =
1 + i
i° =
- 1
1 + g
T
Rt
a
t
t = 0 (1 + i*)
x* = x1 + (x2 - x1) c
Total equity
• Inventory turnover =
T
Linear Interpolation:
Payback Period:
Financial Ratios:
• Equity ratio =
495
1 + i
- 1
1 + f
Real IRR:
IRRR =
1 + IRRC
- 1
1 + f
Simple Interest Amount:
Is = PiN
07/10/20 3:28 PM