Vol II fundamentals of corporate finance, 6th ed

Finance
Fundamentals of Corporate Finance

Volume 2

David Whitehurst
UMIST

abc

McGraw-Hill/Irwin

McGraw−Hill Primis
ISBN: 0−390−31999−6
Text:
Fundamentals of Corporate Finance, Sixth
Edition, Alternate Edition
Ross et al.


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ISBN: 0−390−31999−6


Finance

Volume 2
Ross et al. • Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition
Back Matter

917

Appendix A: Mathematical Tables
Appendix B: Key Equations
Appendix C: Answers to Selected End−of−Chapter Problems
Names Index
Equation Index
Subject Index

917
927
932
938
940

942

iii



Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

Appendix A: Mathematical
Tables

© The McGraw−Hill
Companies, 2002

917

APPENDIX

Mathematical Tables

A

Table A.1
Future value of $1 at the end of t periods ϭ (1 ϩ r)t
Table A.2
Present value of $1 to be received after t periods ϭ 1/(1 ϩ r)t

Table A.3
Present value of an annuity of $1 per period for t periods ϭ [1 Ϫ 1/(1 ϩ r)t]/r
Table A.4
Future value of an annuity of $1 per period for t periods ϭ [(1 ϩ r)t Ϫ 1]/r
Table A.5
Cumulative normal distribution

A-1


918

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-2

TABLE A.1

Future value of $1 at the end of t periods ‫( ؍‬1 ؉ r) t
Interest Rate

Period

1%

2%

3%

4%

5%

6%

7%

8%

TABLE A.1FUTURE VALUE OF $1 AT THE END OF T PERIODS 5 (1 1 R)T

9%

1
2
3
4
5

1.0100
1.0201
1.0303
1.0406
1.0510

1.0200
1.0404
1.0612
1.0824
1.1041

1.0300
1.0609
1.0927
1.1255
1.1593

1.0400
1.0816
1.1249
1.1699
1.2167

1.0500
1.1025
1.1576
1.2155
1.2763

1.0600
1.1236
1.1910
1.2625
1.3382

1.0700
1.1449
1.2250
1.3108
1.4026

1.0800
1.1664
1.2597
1.3605
1.4693

1.0900
1.1881
1.2950
1.4116
1.5386

6
7
8
9
10

1.0615
1.0721
1.0829
1.0937
1.1046

1.1262
1.1487
1.1717
1.1951
1.2190

1.1941
1.2299
1.2668
1.3048
1.3439

1.2653
1.3159
1.3686
1.4233
1.4802

1.3401
1.4071
1.4775
1.5513
1.6289

1.4185
1.5036
1.5938
1.6895
1.7908

1.5007
1.6058
1.7182
1.8385
1.9672

1.5869
1.7138
1.8509
1.9990
2.1589

1.6771
1.8280
1.9926
2.1719
2.3674

11
12
13
14
15

1.1157
1.1268
1.1381
1.1495
1.1610

1.2434
1.2682
1.2936
1.3195
1.3459

1.3842
1.4258
1.4685
1.5126
1.5580

1.5395
1.6010
1.6651
1.7317
1.8009

1.7103
1.7959
1.8856
1.9799
2.0789

1.8983
2.0122
2.1329
2.2609
2.3966

2.1049
2.2522
2.4098
2.5785
2.7590

2.3316
2.5182
2.7196
2.9372
3.1722

2.5804
2.8127
3.0658
3.3417
3.6425

16
17
18
19
20

1.1726
1.1843
1.1961
1.2081
1.2202

1.3728
1.4002
1.4282
1.4568
1.4859

1.6047
1.6528
1.7024
1.7535
1.8061

1.8730
1.9479
2.0258
2.1068
2.1911

2.1829
2.2920
2.4066
2.5270
2.6533

2.5404
2.6928
2.8543
3.0256
3.2071

2.9522
3.1588
3.3799
3.6165
3.8697

3.4259
3.7000
3.9960
4.3157
4.6610

3.9703
4.3276
4.7171
5.1417
5.6044

21
22
23
24
25

1.2324
1.2447
1.2572
1.2697
1.2824

1.5157
1.5460
1.5769
1.6084
1.6406

1.8603
1.9161
1.9736
2.0328
2.0938

2.2788
2.3699
2.4647
2.5633
2.6658

2.7860
2.9253
3.0715
3.2251
3.3864

3.3996
3.6035
3.8197
4.0489
4.2919

4.1406
4.4304
4.7405
5.0724
5.4274

5.0338
5.4365
5.8715
6.3412
6.8485

6.1088
6.6586
7.2579
7.9111
8.6231

30
40
50
60

1.3478
1.4889
1.6446
1.8167

1.8114
2.2080
2.6916
3.2810

2.4273
3.2620
4.3839
5.8916

3.2434
4.8010
7.1067
10.520

4.3219
7.0400
11.467
18.679

5.7435
10.286
18.420
32.988

7.6123
14.974
29.457
57.946

10.063
21.725
46.902
101.26

13.268
31.409
74.358
176.03

Continued on next page


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

919

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical

Tables

APPENDIX A Mathematical Tables

A-3

10%

12%

14%

15%

16%

18%

20%

24%

28%

32%

36%

1.1000
1.2100

1.3310
1.4641
1.6105

1.1200
1.2544
1.4049
1.5735
1.7623

1.1400
1.2996
1.4815
1.6890
1.9254

1.1500
1.3225
1.5209
1.7490
2.0114

1.1600
1.3456
1.5609
1.8106
2.1003

1.1800
1.3924

1.6430
1.9388
2.2878

1.2000
1.4400
1.7280
2.0736
2.4883

1.2400
1.5376
1.9066
2.3642
2.9316

1.2800
1.6384
2.0972
2.6844
3.4360

1.3200
1.7424
2.3000
3.0360
4.0075

1.3600
1.8496

2.5155
3.4210
4.6526

1.7716
1.9487
2.1436
2.3579
2.5937

1.9738
2.2107
2.4760
2.7731
3.1058

2.1950
2.5023
2.8526
3.2519
3.7072

2.3131
2.6600
3.0590
3.5179
4.0456

2.4364
2.8262

3.2784
3.8030
4.4114

2.6996
3.1855
3.7589
4.4355
5.2338

2.9860
3.5832
4.2998
5.1598
6.1917

3.6352
4.5077
5.5895
6.9310
8.5944

4.3980
5.6295
7.2058
9.2234
11.806

5.2899
6.9826

9.2170
12.166
16.060

6.3275
8.6054
11.703
15.917
21.647

2.8531
3.1384
3.4523
3.7975
4.1772

3.4785
3.8960
4.3635
4.8871
5.4736

4.2262
4.8179
5.4924
6.2613
7.1379

4.6524
5.3503

6.1528
7.0757
8.1371

5.1173
5.9360
6.8858
7.9875
9.2655

6.1759
7.2876
8.5994
10.147
11.974

7.4301
8.9161
10.699
12.839
15.407

10.657
13.215
16.386
20.319
25.196

15.112
19.343

24.759
31.691
40.565

21.199
27.983
36.937
48.757
64.359

29.439
40.037
54.451
74.053
100.71

4.5950
5.0545
5.5599
6.1159
6.7275

6.1304
6.8660
7.6900
8.6128
9.6463

8.1372
9.2765

10.575
12.056
13.743

9.3576
10.761
12.375
14.232
16.367

10.748
12.468
14.463
16.777
19.461

14.129
16.672
19.673
23.214
27.393

18.488
22.186
26.623
31.948
38.338

31.243
38.741

48.039
59.568
73.864

51.923
66.461
85.071
108.89
139.38

84.954
112.14
148.02
195.39
257.92

136.97
186.28
253.34
344.54
468.57

7.4002
8.1403
8.9543
9.8497
10.835

10.804
12.100

13.552
15.179
17.000

15.668
17.861
20.362
23.212
26.462

18.822
21.645
24.891
28.625
32.919

22.574
26.186
30.376
35.236
40.874

32.324
38.142
45.008
53.109
62.669

46.005
55.206

66.247
79.497
95.396

91.592
113.57
140.83
174.63
216.54

178.41
228.36
292.30
374.14
478.90

340.45
449.39
593.20
783.02
1033.6

637.26
866.67
1178.7
1603.0
2180.1

17.449
45.259

117.39
304.48

29.960
93.051
289.00
897.60

50.950
188.88
700.23
2595.9

66.212
267.86
1083.7
4384.0

85.850
378.72
1670.7
7370.2

143.37
750.38
3927.4
20555.

237.38
1469.8

9100.4
56348.

634.82
5455.9
46890.
*

1645.5
19427.
*
*

4142.1
66521.
*
*

10143.
*
*
*

*The factor is greater than 99,999.


920

Ross et al.: Fundamentals
of Corporate Finance, Sixth

Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-4

TABLE A.2

Present value of $1 to be received after t periods ‫ ؍‬1/(1 ؉ r) t
Interest Rate

Period

1%

2%

3%

4%

5%

6%

7%

8%

9%

1
2
3
4
5

0.9901
0.9803
0.9706
0.9610
0.9515

0.9804
0.9612
0.9423
0.9238
0.9057

0.9709
0.9426
0.9151

0.8885
0.8626

0.9615
0.9246
0.8890
0.8548
0.8219

0.9524
0.9070
0.8638
0.8227
0.7835

0.9434
0.8900
0.8396
0.7921
0.7473

0.9346
0.8734
0.8163
0.7629
0.7130

0.9259
0.8573
0.7938

0.7350
0.6806

0.9174
0.8417
0.7722
0.7084
0.6499

6
7
8
9
10

0.9420
0.9327
0.9235
0.9143
0.9053

0.8880
0.8706
0.8535
0.8368
0.8203

0.8375
0.8131
0.7894

0.7664
0.7441

0.7903
0.7599
0.7307
0.7026
0.6756

0.7462
0.7107
0.6768
0.6446
0.6139

0.7050
0.6651
0.6274
0.5919
0.5584

0.6663
0.6227
0.5820
0.5439
0.5083

0.6302
0.5835
0.5403

0.5002
0.4632

0.5963
0.5470
0.5019
0.4604
0.4224

11
12
13
14
15

0.8963
0.8874
0.8787
0.8700
0.8613

0.8043
0.7885
0.7730
0.7579
0.7430

0.7224
0.7014
0.6810

0.6611
0.6419

0.6496
0.6246
0.6006
0.5775
0.5553

0.5847
0.5568
0.5303
0.5051
0.4810

0.5268
0.4970
0.4688
0.4423
0.4173

0.4751
0.4440
0.4150
0.3878
0.3624

0.4289
0.3971
0.3677

0.3405
0.3152

0.3875
0.3555
0.3262
0.2992
0.2745

16
17
18
19
20

0.8528
0.8444
0.8360
0.8277
0.8195

0.7284
0.7142
0.7002
0.6864
0.6730

0.6232
0.6050
0.5874

0.5703
0.5537

0.5339
0.5134
0.4936
0.4746
0.4564

0.4581
0.4363
0.4155
0.3957
0.3769

0.3936
0.3714
0.3503
0.3305
0.3118

0.3387
0.3166
0.2959
0.2765
0.2584

0.2919
0.2703
0.2502

0.2317
0.2145

0.2519
0.2311
0.2120
0.1945
0.1784

21
22
23
24
25

0.8114
0.8034
0.7954
0.7876
0.7798

0.6598
0.6468
0.6342
0.6217
0.6095

0.5375
0.5219
0.5067

0.4919
0.4776

0.4388
0.4220
0.4057
0.3901
0.3751

0.3589
0.3418
0.3256
0.3101
0.2953

0.2942
0.2775
0.2618
0.2470
0.2330

0.2415
0.2257
0.2109
0.1971
0.1842

0.1987
0.1839
0.1703

0.1577
0.1460

0.1637
0.1502
0.1378
0.1264
0.1160

30
40
50

0.7419
0.6717
0.6080

0.5521
0.4529
0.3715

0.4120
0.3066
0.2281

0.3083
0.2083
0.1407

0.2314

0.1420
0.0872

0.1741
0.0972
0.0543

0.1314
0.0668
0.0339

0.0994
0.0460
0.0213

0.0754
0.0318
0.0134

Continued on next page


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

921

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-5

10%

12%

14%

15%

16%

18%

20%

24%

28%

32%

36%

0.9091
0.8264
0.7513
0.6830
0.6209

0.8929
0.7972
0.7118
0.6355
0.5674

0.8772
0.7695
0.6750
0.5921
0.5194

0.8696
0.7561
0.6575
0.5718
0.4972

0.8621
0.7432
0.6407
0.5523

0.4761

0.8475
0.7182
0.6086
0.5158
0.4371

0.8333
0.6944
0.5787
0.4823
0.4019

0.8065
0.6504
0.5245
0.4230
0.3411

0.7813
0.6104
0.4768
0.3725
0.2910

0.7576
0.5739
0.4348
0.3294

0.2495

0.7353
0.5407
0.3975
0.2923
0.2149

0.5645
0.5132
0.4665
0.4241
0.3855

0.5066
0.4523
0.4039
0.3606
0.3220

0.4556
0.3996
0.3506
0.3075
0.2697

0.4323
0.3759
0.3269
0.2843

0.2472

0.4104
0.3538
0.3050
0.2630
0.2267

0.3704
0.3139
0.2660
0.2255
0.1911

0.3349
0.2791
0.2326
0.1938
0.1615

0.2751
0.2218
0.1789
0.1443
0.1164

0.2274
0.1776
0.1388
0.1084

0.0847

0.1890
0.1432
0.1085
0.0822
0.0623

0.1580
0.1162
0.0854
0.0628
0.0462

0.3505
0.3186
0.2897
0.2633
0.2394

0.2875
0.2567
0.2292
0.2046
0.1827

0.2366
0.2076
0.1821
0.1597

0.1401

0.2149
0.1869
0.1625
0.1413
0.1229

0.1954
0.1685
0.1452
0.1252
0.1079

0.1619
0.1372
0.1163
0.0985
0.0835

0.1346
0.1122
0.0935
0.0779
0.0649

0.0938
0.0757
0.0610
0.0492

0.0397

0.0662
0.0517
0.0404
0.0316
0.0247

0.0472
0.0357
0.0271
0.0205
0.0155

0.0340
0.0250
0.0184
0.0135
0.0099

0.2176
0.1978
0.1799
0.1635
0.1486

0.1631
0.1456
0.1300
0.1161

0.1037

0.1229
0.1078
0.0946
0.0829
0.0728

0.1069
0.0929
0.0808
0.0703
0.0611

0.0930
0.0802
0.0691
0.0596
0.0514

0.0708
0.0600
0.0508
0.0431
0.0365

0.0541
0.0451
0.0376
0.0313

0.0261

0.0320
0.0258
0.0208
0.0168
0.0135

0.0193
0.0150
0.0118
0.0092
0.0072

0.0118
0.0089
0.0068
0.0051
0.0039

0.0073
0.0054
0.0039
0.0029
0.0021

0.1351
0.1228
0.1117
0.1015

0.0923

0.0926
0.0826
0.0738
0.0659
0.0588

0.0638
0.0560
0.0491
0.0431
0.0378

0.0531
0.0462
0.0402
0.0349
0.0304

0.0443
0.0382
0.0329
0.0284
0.0245

0.0309
0.0262
0.0222
0.0188

0.0160

0.0217
0.0181
0.0151
0.0126
0.0105

0.0109
0.0088
0.0071
0.0057
0.0046

0.0056
0.0044
0.0034
0.0027
0.0021

0.0029
0.0022
0.0017
0.0013
0.0010

0.0016
0.0012
0.0008
0.0006

0.0005

0.0573
0.0221
0.0085

0.0334
0.0107
0.0035

0.0196
0.0053
0.0014

0.0151
0.0037
0.0009

0.0116
0.0026
0.0006

0.0070
0.0013
0.0003

0.0042
0.0007
0.0001

0.0016
0.0002
*

0.0006
0.0001
*

0.0002
*
*

0.0001
*
*

*The factor is zero to four decimal places.


922

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-6

TABLE A.3

Present value of an annuity of $1 per period for t periods ‫[ ؍‬1 ؊ 1/(1 ؉ r) t ]/r
Interest Rate

Number of
Periods

1%

2%

3%

4%

5%

6%

7%

8%

9%

1
2
3
4
5

0.9901
1.9704
2.9410
3.9020
4.8534

0.9804
1.9416
2.8839
3.8077
4.7135

0.9709
1.9135
2.8286
3.7171
4.5797

0.9615
1.8861
2.7751

3.6299
4.4518

0.9524
1.8594
2.7232
3.5460
4.3295

0.9434
1.8334
2.6730
3.4651
4.2124

0.9346
1.8080
2.6243
3.3872
4.1002

0.9259
1.7833
2.5771
3.3121
3.9927

0.9174
1.7591
2.5313

3.2397
3.8897

6
7
8
9
10

5.7955
6.7282
7.6517
8.5660
9.4713

5.6014
6.4720
7.3255
8.1622
8.9826

5.4172
6.2303
7.0197
7.7861
8.5302

5.2421
6.0021
6.7327

7.4353
8.1109

5.0757
5.7864
6.4632
7.1078
7.7217

4.9173
5.5824
6.2098
6.8017
7.3601

4.7665
5.3893
5.9713
6.5152
7.0236

4.6229
5.2064
5.7466
6.2469
6.7101

4.4859
5.0330
5.5348

5.9952
6.4177

11
12
13
14
15

10.3676
11.2551
12.1337
13.0037
13.8651

9.7868
10.5753
11.3484
12.1062
12.8493

9.2526
9.9540
10.6350
11.2961
11.9379

8.7605
9.3851
9.9856

10.5631
11.1184

8.3064
8.8633
9.3936
9.8986
10.3797

7.8869
8.3838
8.8527
9.2950
9.7122

7.4987
7.9427
8.3577
8.7455
9.1079

7.1390
7.5361
7.9038
8.2442
8.5595

6.8052
7.1607
7.4869

7.7862
8.0607

16
17
18
19
20

14.7179
15.5623
16.3983
17.2260
18.0456

13.5777
14.2919
14.9920
15.6785
16.3514

12.5611
13.1661
13.7535
14.3238
14.8775

11.6523
12.1657
12.6593

13.1339
13.5903

10.8378
11.2741
11.6896
12.0853
12.4622

10.1059
10.4773
10.8276
11.1581
11.4699

9.4466
9.7632
10.0591
10.3356
10.5940

8.8514
9.1216
9.3719
9.6036
9.8181

8.3126
8.5436
8.7556

8.9501
9.1285

21
22
23
24
25

18.8570
19.6604
20.4558
21.2434
22.0232

17.0112
17.6580
18.2922
18.9139
19.5235

15.4150
15.9369
16.4436
16.9355
17.4131

14.0292
14.4511
14.8568

15.2470
15.6221

12.8212
13.1630
13.4886
13.7986
14.0939

11.7641
12.0416
12.3034
12.5504
12.7834

10.8355
11.0612
11.2722
11.4693
11.6536

10.0168
10.2007
10.3741
10.5288
10.6748

9.2922
9.4424
9.5802

9.7066
9.8226

30
40
50

25.8077
32.8347
39.1961

22.3965
27.3555
31.4236

19.6004
23.1148
25.7298

17.2920
19.7928
21.4822

15.3725
17.1591
18.2559

13.7648
15.0463
15.7619

12.4090
13.3317
13.8007

11.2578
11.9246
12.2335

10.2737
10.7574
10.9617

Continued on next page


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

923

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-7

10%

12%

14%

15%

16%

18%

20%

24%

28%

32%

36%

0.9091
1.7355
2.4869
3.1699

3.7908

0.8929
1.6901
2.4018
3.0373
3.6048

0.8772
1.6467
2.3216
2.9137
3.4331

0.8696
1.6257
2.2832
2.8550
3.3522

0.8621
1.6052
2.2459
2.7982
3.2743

0.8475
1.5656
2.1743
2.6901

3.1272

0.8333
1.5278
2.1065
2.5887
2.9906

0.8065
1.4568
1.9813
2.4043
2.7454

0.7813
1.3916
1.8684
2.2410
2.5320

0.7576
1.3315
1.7663
2.0957
2.3452

0.7353
1.2760
1.6735
1.9658

2.1807

4.3553
4.8684
5.3349
5.7590
6.1446

4.1114
4.5638
4.9676
5.3282
5.6502

3.8887
4.2883
4.6389
4.9464
5.2161

3.7845
4.1604
4.4873
4.7716
5.0188

3.6847
4.0386
4.3436
4.6065

4.8332

3.4976
3.8115
4.0776
4.3030
4.4941

3.3255
3.6046
3.8372
4.0310
4.1925

3.0205
3.2423
3.4212
3.5655
3.6819

2.7594
2.9370
3.0758
3.1842
3.2689

2.5342
2.6775
2.7860
2.8681

2.9304

2.3388
2.4550
2.5404
2.6033
2.6495

6.4951
6.8137
7.1034
7.3667
7.6061

5.9377
6.1944
6.4235
6.6282
6.8109

5.4527
5.6603
5.8424
6.0021
6.1422

5.2337
5.4206
5.5831
5.7245

5.8474

5.0286
5.1971
5.3423
5.4675
5.5755

4.6560
4.7932
4.9095
5.0081
5.0916

4.3271
4.4392
4.5327
4.6106
4.6755

3.7757
3.8514
3.9124
3.9616
4.0013

3.3351
3.3868
3.4272
3.4587

3.4834

2.9776
3.0133
3.0404
3.0609
3.0764

2.6834
2.7084
2.7268
2.7403
2.7502

7.8237
8.0216
8.2014
8.3649
8.5136

6.9740
7.1196
7.2497
7.3658
7.4694

6.2651
6.3729
6.4674
6.5504

6.6231

5.9542
6.0472
6.1280
6.1982
6.2593

5.6685
5.7487
5.8178
5.8775
5.9288

5.1624
5.2223
5.2732
5.3162
5.3527

4.7296
4.7746
4.8122
4.8435
4.8696

4.0333
4.0591
4.0799
4.0967

4.1103

3.5026
3.5177
3.5294
3.5386
3.5458

3.0882
3.0971
3.1039
3.1090
3.1129

2.7575
2.7629
2.7668
2.7697
2.7718

8.6487
8.7715
8.8832
8.9847
9.0770

7.5620
7.6446
7.7184
7.7843

7.8431

6.6870
6.7429
6.7921
6.8351
6.8729

6.3125
6.3587
6.3988
6.4338
6.4641

5.9731
6.0113
6.0442
6.0726
6.0971

5.3837
5.4099
5.4321
5.4509
5.4669

4.8913
4.9094
4.9245
4.9371

4.9476

4.1212
4.1300
4.1371
4.1428
4.1474

3.5514
3.5558
3.5592
3.5619
3.5640

3.1158
3.1180
3.1197
3.1210
3.1220

2.7734
2.7746
2.7754
2.7760
2.7765

9.4269
9.7791
9.9148

8.0552
8.2438
8.3045

7.0027
7.1050
7.1327

6.5660
6.6418
6.6605

6.1772
6.2335
6.2463

5.5168
5.5482
5.5541

4.9789
4.9966
4.9995

4.1601
4.1659
4.1666

3.5693
3.5712

3.5714

3.1242
3.1250
3.1250

2.7775
2.7778
2.7778


924

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-8

Future value of an annuity of $1 per period for t periods ‫([ ؍‬1 ؉ r) t ؊ 1]/r

TABLE A.4

Interest Rate
Number of
Periods

1%

2%

3%

4%

5%

6%

7%

8%

9%

1
2
3
4
5

1.0000
2.0100
3.0301
4.0604
5.1010

1.0000
2.0200
3.0604
4.1216
5.2040

1.0000
2.0300
3.0909
4.1836
5.3091

1.0000
2.0400
3.1216
4.2465
5.4163

1.0000
2.0500
3.1525
4.3101
5.5256

1.0000
2.0600
3.1836
4.3746
5.6371

1.0000
2.0700
3.2149
4.4399
5.7507

1.0000
2.0800
3.2464
4.5061
5.8666

1.0000
2.0900
3.2781
4.5731
5.9847

6
7
8
9
10

6.1520
7.2135
8.2857
9.3685
10.462

6.3081
7.4343
8.5830
9.7546
10.950

6.4684
7.6625
8.8932
10.159
11.464

6.6330
7.8983
9.2142
10.583
12.006

6.8019
8.1420
9.5491
11.027
12.578

6.9753
8.3938
9.8975
11.491
13.181

7.1533
8.6540
10.260
11.978
13.816

7.3359
8.9228
10.637
12.488
14.487

7.5233
9.2004
11.028
13.021
15.193

11
12
13
14
15

11.567
12.683
13.809
14.947
16.097

12.169
13.412
14.680
15.974
17.293

12.808
14.192
15.618
17.086
18.599

13.486
15.026
16.627
18.292
20.024

14.207
15.917
17.713
19.599
21.579

14.972
16.870
18.882
21.015
23.276

15.784
17.888
20.141
22.550
25.129

16.645
18.977
21.495
24.215
27.152

17.560
20.141
22.953
26.019
29.361

16
17
18
19
20

17.258
18.430
19.615
20.811
22.019

18.639
20.012
21.412
22.841
24.297

20.157
21.762
23.414
25.117
26.870

21.825
23.698
25.645
27.671
29.778

23.657
25.840
28.132
30.539
33.066

25.673
28.213
30.906
33.760
36.786

27.888
30.840
33.999
37.379
40.995

30.324
33.750
37.450
41.446
45.762

33.003
36.974
41.301
46.018
51.160

21
22
23
24
25

23.239
24.472
25.716
26.973
28.243

25.783
27.299
28.845
30.422
32.030

28.676
30.537
32.453
34.426
36.459

31.969
34.248
36.618
39.083
41.646

35.719
38.505
41.430
44.502
47.727

39.993
43.392
46.996
50.816
54.865

44.865
49.006
53.436
58.177
63.249

50.423
55.457
60.893
66.765
73.106

56.765
62.873
69.532
76.790
84.701

30
40
50
60

34.785
48.886
64.463
81.670

40.568
60.402
84.579
114.05

47.575
75.401
112.80
163.05

56.085
95.026
152.67
237.99

66.439
120.80
209.35
353.58

79.058
154.76
290.34
533.13

94.461
199.64
406.53
813.52

113.28
259.06
573.77
1253.2

136.31
337.88
815.08
1944.8

Continued on next page


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

925

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical

Tables

APPENDIX A Mathematical Tables

A-9

10%

12%

14%

15%

16%

18%

20%

24%

28%

32%

36%

1.0000
2.1000

3.3100
4.6410
6.1051

1.0000
2.1200
3.3744
4.7793
6.3528

1.0000
2.1400
3.4396
4.9211
6.6101

1.0000
2.1500
3.4725
4.9934
6.7424

1.0000
2.1600
3.5056
5.0665
6.8771

1.0000
2.1800

3.5724
5.2154
7.1542

1.0000
2.2000
3.6400
5.3680
7.4416

1.0000
2.2400
3.7776
5.6842
8.0484

1.0000
2.2800
3.9184
6.0156
8.6999

1.0000
2.3200
4.0624
6.3624
9.3983

1.0000
2.3600

4.2096
6.7251
10.146

7.7156
9.4872
11.436
13.579
15.937

8.1152
10.089
12.300
14.776
17.549

8.5355
10.730
13.233
16.085
19.337

8.7537
11.067
13.727
16.786
20.304

8.9775
11.414

14.240
17.519
21.321

9.4420
12.142
15.327
19.086
23.521

9.9299
12.916
16.499
20.799
25.959

10.980
14.615
19.123
24.712
31.643

12.136
16.534
22.163
29.369
38.593

13.406
18.696

25.678
34.895
47.062

14.799
21.126
29.732
41.435
57.352

18.531
21.384
24.523
27.975
31.772

20.655
24.133
28.029
32.393
37.280

23.045
27.271
32.089
37.581
43.842

24.349
29.002

34.352
40.505
47.580

25.733
30.850
36.786
43.672
51.660

28.755
34.931
42.219
50.818
60.965

32.150
39.581
48.497
59.196
72.035

40.238
50.895
64.110
80.496
100.82

50.398
65.510

84.853
109.61
141.30

63.122
84.320
112.30
149.24
198.00

78.998
108.44
148.47
202.93
276.98

35.950
40.545
45.599
51.159
57.275

42.753
48.884
55.750
63.440
72.052

50.980
59.118

68.394
78.969
91.025

55.717
65.075
75.836
88.212
102.44

60.925
71.673
84.141
98.603
115.38

72.939
87.068
103.74
123.41
146.63

87.442
105.93
128.12
154.74
186.69

126.01
157.25

195.99
244.03
303.60

181.87
233.79
300.25
385.32
494.21

262.36
347.31
459.45
607.47
802.86

377.69
514.66
700.94
954.28
1298.8

64.002
71.403
79.543
88.497
98.347

81.699
92.503

104.60
118.16
133.33

104.77
120.44
138.30
158.66
181.87

118.81
137.63
159.28
184.17
212.79

134.84
157.41
183.60
213.98
249.21

174.02
206.34
244.49
289.49
342.60

225.03
271.03

326.24
392.48
471.98

377.46
469.06
582.63
723.46
898.09

633.59
812.00
1040.4
1332.7
1706.8

1060.8
1401.2
1850.6
2443.8
3226.8

1767.4
2404.7
3271.3
4450.0
6053.0

164.49
442.59

1163.9
3043.8

241.33
767.09
2400.0
7471.6

356.79
1342.0
4994.5
18535.

434.75
1779.1
7217.7
29220.

530.31
2360.8
10436.
46058.

790.95
4163.2
21813.
*

1181.9
7343.9

45497.
*

2640.9
22729.
*
*

5873.2
69377.
*
*

12941.
*
*
*

28172.
*
*
*

*The factor is greater than 99,999.


926

Ross et al.: Fundamentals
of Corporate Finance, Sixth

Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix A: Mathematical
Tables

APPENDIX A Mathematical Tables

A-10

TABLE A.5

Cumulative normal distribution

d

N(d)

d

N(d)

d

N(d)

d

N(d)

d

N(d)

d

N(d)

؊3.00
؊2.95
؊2.90
؊2.85
؊2.80
؊2.75
؊2.70
؊2.65
؊2.60
؊2.55
؊2.50
؊2.45
؊2.40
؊2.35
؊2.30
؊2.25
؊2.20
؊2.15

؊2.10
؊2.05
؊2.00
؊1.98
؊1.96
؊1.94
؊1.92
؊1.90
؊1.88
؊1.86
؊1.84
؊1.82
؊1.80
؊1.78
؊1.76
؊1.74
؊1.72
؊1.70
؊1.68
؊1.66
؊1.64
؊1.62
؊1.60

.0013
.0016
.0019
.0022
.0026
.0030

.0035
.0040
.0047
.0054
.0062
.0071
.0082
.0094
.0107
.0122
.0139
.0158
.0179
.0202
.0228
.0239
.0250
.0262
.0274
.0287
.0301
.0314
.0329
.0344
.0359
.0375
.0392
.0409
.0427
.0446

.0465
.0485
.0505
.0526
.0548

؊1.58
؊1.56
؊1.54
؊1.52
؊1.50
؊1.48
؊1.46
؊1.44
؊1.42
؊1.40
؊1.38
؊1.36
؊1.34
؊1.32
؊1.30
؊1.28
؊1.26
؊1.24
؊1.22
؊1.20
؊1.18
؊1.16
؊1.14
؊1.12

؊1.10
؊1.08
؊1.06
؊1.04
؊1.02
؊1.00
؊0.98
؊0.96
؊0.94
؊0.92
؊0.90
؊0.88
؊0.86
؊0.84
؊0.82
؊0.80
؊0.78

.0571
.0594
.0618
.0643
.0668
.0694
.0721
.0749
.0778
.0808
.0838
.0869

.0901
.0934
.0968
.1003
.1038
.1075
.1112
.1151
.1190
.1230
.1271
.1314
.1357
.1401
.1446
.1492
.1539
.1587
.1635
.1685
.1736
.1788
.1841
.1894
.1949
.2005
.2061
.2119
.2177

؊0.76
؊0.74
؊0.72
؊0.70
؊0.68
؊0.66
؊0.64
؊0.62
؊0.60
؊0.58
؊0.56
؊0.54
؊0.52
؊0.50
؊0.48
؊0.46
؊0.44
؊0.42
؊0.40
؊0.38
؊0.36
؊0.34
؊0.32
؊0.30
؊0.28
؊0.26
؊0.24
؊0.22
؊0.20
؊0.18

؊0.16
؊0.14
؊0.12
؊0.10
؊0.08
؊0.06
؊0.04
؊0.02
0.00
0.02
0.04

.2236
.2297
.2358
.2420
.2483
.2546
.2611
.2676
.2743
.2810
.2877
.2946
.3015
.3085
.3156
.3228
.3300
.3373

.3446
.3520
.3594
.3669
.3745
.3821
.3897
.3974
.4052
.4129
.4207
.4286
.4365
.4443
.4523
.4602
.4681
.4761
.4841
.4920
.5000
.5080
.5160

0.06
0.08
0.10
0.12
0.14
0.16

0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76

0.78
0.80
0.82
0.84

.5239
.5319
.5398
.5478
.5557
.5636
.5714
.5793
.5871
.5948
.6026
.6103
.6179
.6255
.6331
.6406
.6480
.6554
.6628
.6700
.6773
.6844
.6915
.6985
.7054

.7123
.7191
.7258
.7324
.7389
.7454
.7518
.7580
.7642
.7704
.7764
.7823
.7882
.7939
.7996

0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12

1.14
1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.30
1.32
1.34
1.36
1.38
1.40
1.42
1.44
1.46
1.48
1.50
1.52
1.54
1.56
1.58
1.60
1.62
1.64

.8051
.8106
.8159

.8212
.8264
.8315
.8365
.8414
.8461
.8508
.8554
.8599
.8643
.8686
.8729
.8770
.8810
.8849
.8888
.8925
.8962
.8997
.9032
.9066
.9099
.9131
.9162
.9192
.9222
.9251
.9279
.9306
.9332

.9357
.9382
.9406
.9429
.9452
.9474
.9495

1.66
1.68
1.70
1.72
1.74
1.76
1.78
1.80
1.82
1.84
1.86
1.88
1.90
1.92
1.94
1.96
1.98
2.00
2.05
2.10
2.15
2.20

2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.05

.9515
.9535
.9554
.9573
.9591
.9608
.9625
.9641
.9656
.9671
.9686
.9699

.9713
.9726
.9738
.9750
.9761
.9772
.9798
.9821
.9842
.9861
.9878
.9893
.9906
.9918
.9929
.9938
.9946
.9953
.9960
.9965
.9970
.9974
.9978
.9981
.9984
.9986
.9989

This table shows the probability [N(d)] of observing a value less than or equal to d. For example, as illustrated, if d is ؊.24, then N(d) is .4052.

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix B: Key Equations

927

APPENDIX

B

Key Equations
Chapter 2
1

2
3

The balance sheet identity or equation:
Assets ϭ Liabilities
[2.1]
ϩ Shareholders’ equity
The income statement equation:

Revenues Ϫ Expenses ϭ Income
[2.2]
The cash flow identity:
Cash flow from assets
ϭ Cash flow to creditors
[2.3]
ϩ Cash flow to stockholders
where
a. Cash flow from assets ϭ Operating cash flow
(OCF) Ϫ Net capital spending Ϫ Change in net
working capital (NWC)
(1) Operating cash flow ϭ Earnings before
interest and taxes (EBIT) ϩ Depreciation
Ϫ Taxes
(2) Net capital spending ϭ Ending net fixed
assets Ϫ Beginning net fixed assets
ϩ Depreciation
(3) Change in net working capital ϭ Ending
NWC Ϫ Beginning NWC
b. Cash flow to creditors ϭ Interest paid Ϫ Net new
borrowing
c. Cash flow to stockholders ϭ Dividends paid
Ϫ Net new equity raised

4

The ratio of net working capital to total assets:
Net working capital to total assets
Net working capital
ϭ

[3.4]
Total assets

5

The interval measure:
Interval measure
Current assets
ϭ
Average daily operating costs

6

7

8

9

10

Chapter 3
1

The current ratio:
Current assets
Current liabilities
The quick or acid-test ratio:
Current assets Ϫ Inventory
Quick ratio ϭ

Current liabilities
The cash ratio:
Cash
Cash ratio ϭ
Current liabilities
Current ratio ϭ

2

3

11
[3.1]

The total debt ratio:
Total debt ratio
Total assets Ϫ Total equity
ϭ
Total assets
The debt-equity ratio:
Debt-equity ratio
ϭ Total debt/Total equity
The equity multiplier:
Equity multiplier
ϭ Total assets/Total equity
The long-term debt ratio:
Long-term debt ratio
Long-term debt
ϭ
Long-term debt ϩ Total equity

The times interest earned (TIE) ratio:
EBIT
Times interest earned ratio ϭ
Interest

[3.3]

12

[3.6]

[3.7]

[3.8]

[3.9]

[3.10]

The cash coverage ratio:
Cash coverage ratio
ϭ

[3.2]

[3.5]

EBIT ϩ Depreciation
Interest

The inventory turnover ratio:
Inventory turnover
Cost of goods sold
ϭ
Inventory

[3.11]

[3.12]
B-1


928

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

14

15

16

17

18

19

20

21

22

23

The average days’ sales in inventory:
Days’ sales in inventory
365 days
ϭ
[3.13]
Inventory turnover
The receivables turnover ratio:
Receivables turnover
Sales
ϭ
[3.14]
Accounts receivable
The days’ sales in receivables:
Days’ sales in receivables
365 days
ϭ
[3.15]
Receivables turnover
The net working capital (NWC) turnover ratio:
Sales
NWC turnover ϭ
[3.16]

NWC
The fixed asset turnover ratio:
Sales
Fixed asset turnover ϭ
[3.17]
Net fixed assets
The total asset turnover ratio:
Sales
Total asset turnover ϭ
[3.18]
Total assets
Profit margin:
Net income
Profit margin ϭ
[3.19]
Sales
Return on assets (ROA):
Net income
Return on assets ϭ
[3.20]
Total assets
Return on equity (ROE):
Net income
Return on equity ϭ
[3.21]
Total equity
The price-earnings (PE) ratio:
Price per share
PE ratio ϭ
[3.22]

Earnings per share
The market-to-book ratio:
Market-to-book ratio
Market value per share
ϭ
[3.23]
Book value per share
The Du Pont identity:
Net income
Sales
Assets
ROE ϭ
ϫ
ϫ
[3.24]
Sales
Assets Equity

2

The internal growth rate:

3

ROA ϫ b
1 Ϫ ROA ϫ b
The sustainable growth rate:
Internal growth rate ϭ

Sustainable growth rate ϭ

4

Return on assets
ROE ϭ Profit margin
ϫ Total asset turnover
ϫ Equity multiplier

ϭ

The dividend payout ratio:
Dividend payout ratio
ϭ Cash dividends/Net income

1

2

3

Total assets
Sales

1
Total asset turnover

The future value of $1 invested for t periods at rate
of r per period:
Future value ϭ $1 ϫ (1 ϩ r)t
[5.1]
The present value of $1 to be received t periods in

the future at a discount rate of r:
PV ϭ $1 ϫ [1/(1 ϩ r)t] ϭ $1/(1 ϩ r)t
[5.2]
The relationship between future value and present
value (the basic present value equation):
PV ϫ (1 ϩ r)t ϭ FVt
[5.3]
PV ϭ FVt /(1 ϩ r)t ϭ FVt ϫ [1/(1 ϩ r)t]

Chapter 6
1

The present value of an annuity of C dollars per
period for t periods when the rate of return or interest
rate is r:
Annuity present value

(1 Ϫ Presentrvalue factor)
1 Ϫ [1/(1 ϩ r) ]
ϭCϫ{
}
r

ϭCϫ

t

2

3

4
5

6
[4.1]

[4.3]

Chapter 5

Chapter 4
1

ROE ϫ b
1 Ϫ ROE ϫ b

[4.2]

The capital intensity ratio:
Capital intensity ratio ϭ

Ά

24

© The McGraw−Hill
Companies, 2002

Appendix B: Key Equations

APPENDIX B Key Equations

B-2

13

Back Matter

[6.1]

The future value factor for an annuity:
Annuity FV factor
ϭ (Future value factor Ϫ 1)/r
[6.2]
ϭ [(1 ϩ r)t Ϫ 1]/r
Annuity due value ϭ Ordinary annuity value
[6.3]
ϫ (1 ϩ r)
Present value for a perpetuity:
PV for a perpetuity ϭ C/r ϭ C ϫ (1/r)
[6.4]
Effective annual rate (EAR), where m is the number
of times the interest is compounded during the year:
EAR ϭ [1 ϩ (Quoted rate/m)]m Ϫ 1
[6.5]
Effective annual rate (EAR), where q stands for the
continuously compounded quoted rate:
EAR ϭ eq Ϫ 1
[6.6]

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix B: Key Equations

APPENDIX B Key Equations

3

Chapter 7
1

2

Bond value if bond has (1) a face value of F paid
at maturity, (2) a coupon of C paid per period,
(3) t periods to maturity, and (4) a yield of r per
period:
Bond value
ϭ C ϫ [1 Ϫ 1/(1 ϩ r)t]/r ϩ F/(1 ϩ r)t
[7.1]
Bond value
Present value

Present value
ϭ
ϩ
of the coupons of the face amount
The Fisher effect:
1 ϩ R ϭ (1 ϩ r) ϫ (1 ϩ h)
[7.2]
Rϭrϩhϩrϫh
[7.3]
RϷrϩh
[7.4]

2

The dividend growth model:
D1
D ϫ (1 ϩ g)
ϭ
P0 ϭ 0
RϪg
RϪg
Required return:
R ϭ D1/P0 ϩ g

1
2

3
4

5
[8.3]

2

3

4

5

6

Net present value (NPV):
NPV ϭ Present value of future cash flows
Ϫ Investment cost
Payback period:
Payback period ϭ Number of years that pass
before the sum of an investment’s cash flows
equals the cost of the investment
Discounted payback period:
Discounted payback period ϭ Number of years
that pass before the sum of an investment’s
discounted cash flows equals the cost of the
investment
The average accounting return (AAR):
Average net income
AAR ϭ
Average book value
Internal rate of return (IRR):

IRR ϭ Discount rate of required return such that
the net present value of an investment is zero
Profitability index:
PV of cash flows
Profitability index ϭ
Cost of investment

Chapter 10
1
2

Tax shield approach to operating cash flow (OCF):
OCF ϭ (Sales Ϫ Costs) ϫ (1 Ϫ T)
[10.3]
ϩ Depreciation ϫ T

Accounting break-even level:
Q ϭ (FC ϩ D)/(P Ϫ v)
[11.1]
Relationship between operating cash flow (OCF) and
sales volume:
Q ϭ (FC ϩ OCF)/(P Ϫ v)
[11.3]
Cash break-even level:
Q ϭ FC/(P Ϫ v)
Financial break-even level:
Q ϭ (FC ϩ OCF*)/(P Ϫ v)
where
OCF* ϭ Zero NPV cash flow
Degree of operating leverage (DOL):

DOL ϭ 1 ϩ FC/OCF
[11.4]

Chapter 12
[8.5]

1

Chapter 9
1

B-3

Chapter 11

Chapter 8
1

929

Bottom-up approach to operating cash flow (OCF):
OCF ϭ Net income ϩ Depreciation
[10.1]
Top-down approach to operating cash flow (OCF):
OCF ϭ Sales Ϫ Costs Ϫ Taxes
[10.2]

2

Variance of returns, Var(R) or ␴ 2 :

1
¯ )2 ϩ . . .
Var(R) ϭ
[(R Ϫ R
TϪ1 1
¯ )2]
ϩ (RT Ϫ R
Standard deviation of returns, SD(R) or ␴:
Var(R)
SD(R) ϭ ͙ළළළළළ

[12.3]

Chapter 13
1

2

3

Risk premium:
Risk premium ϭ Expected return
Ϫ Risk-free rate
Expected return on a portfolio:
E(RP) ϭ x1 ϫ E(R1) ϩ x2 ϫ E(R2 ) ϩ . . .
ϩ xn ϫ E(Rn )
The reward-to-risk ratio:
E[Ri ] Ϫ Rf
␤i
The capital asset pricing model (CAPM):

E(Ri ) ϭ Rf ϩ [E(RM ) Ϫ Rf ] ϫ ␤i

[13.1]

[13.2]

Reward-to-risk ratio ϭ

4

[13.7]

Chapter 14
1

2

Value of a call option at maturity:
a. C1 ϭ 0 if (S1 Ϫ E ) Յ 0
b. C1 ϭ S1 Ϫ E if (S1 Ϫ E ) Ͼ 0
Bounds on the value of a call option:
a. Upper bound:
C0 Յ S0
b. Lower bound:
C0 Ն 0 if S0 Ϫ E Ͻ 0
C0 Ն S0 Ϫ E if S0 Ϫ E Ն 0

[14.1]
[14.2]

[14.3]
[14.4]


930

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

4

S0 ϭ C0 ϩ E/(1 ϩ Rf )
[14.5]
C0 ϭ S0 Ϫ E/(1 ϩ Rf )
Value of a call that is certain to finish in-the-money:
Call option value
ϭ Stock value
Ϫ Present value of the exercise price
C0 ϭ S0 Ϫ E/(1 ϩ Rf )t
[14.6]

2
3
4

5

c. Proposition II:
RE ϭ RU ϩ (RU Ϫ RD) ϫ (D/E)

ϫ (1 Ϫ TC )

Required return on equity, RE (dividend growth
model):
RE ϭ D1 /P0 ϩ g
[15.1]
Required return on equity, RE (CAPM):
RE ϭ Rf ϩ ␤E ϫ (RM Ϫ Rf )
[15.2]
Required return on preferred stock, RP :
RP ϭ D/P0
[15.3]
The weighted average cost of capital (WACC):
WACC ϭ (E/V) ϫ RE ϩ (D/V) ϫ RD
[15.6]
ϫ (1 Ϫ TC )
Weighted average flotation cost, fA :
E
D
[15.8]
fA ϭ ϫ fE ϩ ϫ fD
V
V

1

Rights offerings:
a. Number of new shares:
Number of new shares
Funds to be raised

ϭ
Subscription price
b. Number of rights needed:
Old
Number of rights needed shares
ϭ
to buy a share of stock
New
shares
c. Value of a right:
Value of a right ϭ Rights-on price
Ϫ Ex-rights price

1

2

Modigliani-Miller Propositions (no taxes):
a. Proposition I:
VL ϭ VU
b. Proposition II:
RE ϭ RA ϩ (RA Ϫ RD ) ϫ (D/E)
Modigliani-Miller propositions (with taxes):
a. Value of the interest tax shield:
Value of the interest tax shield
ϭ (TC ϫ RD ϫ D)/RD
ϭ TC ϫ D
b. Proposition I:
VL ϭ VU ϩ TC ϫ D

[19.4]

[19.5]

Float measurement:
a. Average daily float:
Average daily float ϭ

2

[16.1]
3
[16.2]

Total float
Total days

b. Average daily float:
Average daily float
ϭ Average daily receipts
ϫ Weighted average delay
The Baumol-Allais-Tobin (BAT) model:
a. Opportunity costs:
Opportunity costs ϭ (C/2) ϫ R
b. Trading costs:
Trading costs ϭ (T/C ) ϫ F
c. Total cost:
Total cost ϭ Opportunity costs
ϩ Trading costs
d. The optimal initial cash balance:

C* ϭ ͙ළළළළළ
(2T ϫළළ
F)/R
ළළළ
The Miller-Orr model:
a. The optimal cash balance:
C* ϭ L ϩ (3/4 ϫ F ϫ ␴ 2/R)1/3
b. The upper limit:
U* ϭ 3 ϫ C* Ϫ 2 ϫ L

[20.1]

[20.2]

[20A.1]
[20A.2]
[20A.3]
[20A.4]

[20A.5]
[20A.6]

Chapter 21
1

Chapter 17
1

The operating cycle:
Operating cycle ϭ Inventory period

ϩ Accounts receivable period
The cash cycle:
Cash cycle ϭ Operating cycle
Ϫ Accounts payable period

Chapter 20

Chapter 16
1

[17.4]

Chapter 19

2

Chapter 15
1

© The McGraw−Hill
Companies, 2002

Appendix B: Key Equations

APPENDIX B Key Equations

B-4

3

Back Matter

2

[17.1]

[17.2]
3
[17.3]

The size of receivables:
Accounts receivable
[21.1]
ϭ Average daily sales ϫ ACP
NPV of switching credit terms:
a. Present value of switching:
PV ϭ [(P Ϫ v)(QЈ Ϫ Q)]/R
[21.4]
b. Cost of switching:
Cost of switching ϭ PQ ϩ v(QЈ Ϫ Q) [21.5]
c. NPV of switching:
NPV of switching ϭ Ϫ[PQ ϩ v(QЈ Ϫ Q)]
ϩ (P Ϫ v)
[21.6]
ϫ (QЈ Ϫ Q)/R
NPV of granting credit:
a. With no repeat business:
NPV ϭ Ϫv ϩ (1 Ϫ ␲)P/(1 ϩ R)
[21.8]

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix B: Key Equations

APPENDIX B Key Equations

4

b. With repeat business:
NPV ϭ Ϫv ϩ (1 Ϫ ␲)(P Ϫ v)/R
The economic order quantity (EOQ) model:
a. Total carrying costs:
Total carrying costs
ϭ Average inventory
ϫ Carrying costs per unit
ϭ (Q/2) ϫ CC
b. Total restocking costs:
Total restocking costs
ϭ Fixed cost per order
ϫ Number of orders ϭ F ϫ (T/Q)
c. Total costs:
Total costs ϭ Carrying costs

ϩ Restocking costs
ϭ (Q/2) ϫ CC
ϩ F ϫ (T/Q)
d. The optimal order size Q*:
2T ϫ F
Q* ϭ
CC

͙

3
[21.9]
4

[21.10]

2

Purchasing power parity (PPP):
E(St) ϭ S0 ϫ [1 ϩ (hFC Ϫ hUS)]t
Interest rate parity (IRP):
a. Exact, single period:
F1/S0 ϭ (1 ϩ RFC )/(1 ϩ RUS )
b. Approximate, multiperiod:
Ft ϭ S0 ϫ [1 ϩ (RFC Ϫ RUS )]t

2

[21.12]
3

[22.9]
[22.10]

Put-call parity condition:
S ϩ P ϭ PV(E) ϩ C
The Black-Scholes call option formula:
C ϭ S ϫ N(d1) Ϫ E ϫ eϪRt ϫ N(d2)
where
d1 ϭ [1n(S/E) ϩ (R ϩ ␴ 2/2) ϫ t]/(␴ ϫ ͙ළt )
d2 ϭ d1 Ϫ ␴ ϫ ͙ළt
Value of a risk-free bond:
Value of risky bond ϩ put option

[24.2]
[24.5]

[24.6]

[24.7]

[21.16]

Chapter 25
1

Chapter 22
1

B-5

Chapter 24
1

[21.11]

Uncovered interest parity (UIP):
E(St) ϭ S0 ϫ [1 ϩ (RFC Ϫ RUS )]t
International Fisher effect (IFE):
RUS Ϫ hUS ϭ RFC Ϫ hFC

931

[22.3]

[22.4]
[22.7]

The NPV of a merger:
NPV ϭ VB* Ϫ Cost to Firm A of
the acquisition

[25.1]


932

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

Appendix C: Answers to
Selected End−of−Chapter
Problems

© The McGraw−Hill
Companies, 2002

APPENDIX

C

Answers to Selected
End-of-Chapter Problems

Chapter 2
2.2
2.4
2.6
2.8
2.10
2.12
2.14

2.16
2.18

2.20

2.22

2.24

2.26

Net income ϭ $122,850
EPS ϭ $4.10
DPS ϭ $2.17
Taxes ϭ $55,400
OCF ϭ $3,040.50
Change in NWC ϭ $435
Cash flow to stockholders ϭ Ϫ$175,000
a. OCF ϭ $36,170
b. Cash flow to creditors ϭ $20,000
c. Cash flow to stockholders ϭ $3,570
d. Change in NWC ϭ $1,600
Common stock ϭ $855,000
a. Taxgrowth ϭ $15,450
Taxincome ϭ $3,060,000
b. $3,400
Net new long-term debt ϭ Ϫ$20,000
a. Owners’ equity:
2001 ϭ $1,780
2002 ϭ $1,852
b. Change in NWC ϭ Ϫ$28
c. Fixed assets sold ϭ $500
Cash flow from assets ϭ $2,064.20
d. Debt retired ϭ $100
Cash flow to creditors ϭ $12

b. Average tax rate ϭ 34%
Average tax rate ϭ 35%
c. Bubble rate ϭ 45.75%
Cash flow from assets ϭ $215.14
Cash flow to creditors ϭ Ϫ$619.00
Cash flow to stockholders ϭ $834.14

3.4

3.6

3.8
3.10
3.12

3.18
3.20
3.22

3.24
3.26

Chapter 3
3.2

C

Net income ϭ $2.24 million
ROA ϭ 5.21%
ROE ϭ 6.59%

Inventory turnover ϭ 5.59 times
Days’ sales in inventory ϭ 65.35 days
Average inventory period ϭ 65.35 days
EPS ϭ $3.40
DPS ϭ $1.20
BVPS ϭ $48.00
Market-to-book ratio ϭ 1.98 times
PE ratio ϭ 27.9 times
Debt-equity ratio ϭ .75 times
74.18 days
Equity multiplier ϭ 2.10 times
ROE ϭ 17.64%
Net income ϭ $77,616
Net income ϭ $91.80
Net fixed assets ϭ $5,400.91
Profit margin ϭ 5.20%
Total asset turnover ϭ 2.34 times
ROE ϭ 27.97%
TIE ratio ϭ 2.23 times
a. 4.33 times; 3.61 times
b. 1.77 times; 1.30 times
c. .38 times; .33 times
d. 1.06 times
e. 2.33 times
f. 13.30 times
g. .28; .24
h. .39; .32
i. 1.39; 1.32
j. 14.55 times

k. 16.73 times
l. 30.83%
m. 32.72%
n. 43.10%

Chapter 4
4.2

EFN ϭ Ϫ$1,770


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix C: Answers to
Selected End−of−Chapter
Problems

APPENDIX C Answers to Selected End-of-Chapter Problems

4.4
4.6
4.8
4.12

4.14
4.16
4.18
4.20
4.22

4.24
4.26

4.28

EFN ϭ $22,046
Internal growth rate ϭ 4.03%
Maximum increase in sales ϭ $6,163
Internal growth rate ϭ 9.89%
Sustainable growth rate ϭ 9.89%
ROE ϭ 23.00%
Maximum sales growth ϭ 33.33%
Profit margin ϭ 20.43%
TAT ϭ 1.15 times
Sustainable growth rate ϭ 46.79%
New borrowing ϭ $30,413
Internal growth rate ϭ 11.75%
EFN ϭ Ϫ$79,646
EFN @ 20.00% ϭ $12,754
EFN @ 25.00% ϭ $32,732
EFN @ 30.00% ϭ $52,710
EFN @ 16.81% ϭ $0
Maximum sustainable growth rate ϭ 3.73%

6.6
6.8
6.10
6.12
6.14
6.16
6.18
6.20
6.22
6.24
6.26
6.28
6.30

Chapter 5
5.2

5.4

5.6
5.8
5.10
5.12
5.14
5.16

5.18
5.20

$67,410

$36,964
$128,670
$258,619
5.03%
8.67%
8.72%
5.85%
11.77%
8.36%
$127.15 million
$4,547.83
$7.00
a. 10.50%
b. 11.97%
c. 7.62%
$96,654.57
$40,827.94
23.93 years

Chapter 6
6.2

6.4

@ 5%: PVx ϭ $19,389.64
PVy ϭ $17,729.75
@ 22%: PVx ϭ $10,857.80
PVy ϭ $12,468.20
15 years: PV ϭ $31,184.93
40 years: PV ϭ $40,094.11

6.36
6.38
6.40
6.42
6.44
6.46
6.48
6.50
6.52
6.54
6.56
6.58
6.60
6.62
6.64

75 years: PV ϭ$40,967.76
Forever: PV ϭ$41,000.00
PV ϭ $439,297.77
C ϭ $8,834.47
PV ϭ $55,555.56
12.55%; 8.30%; 7.25%; 17.35%
1st National: EAR ϭ 9.49%
1st United: EAR ϭ 9.41%
$5,107.99
$9,249.39
C ϭ $1,020.43
EAR ϭ 10.25%
APR ϭ 1,733.33%

EAR ϭ 313,916,515.70%
FV ϭ $86,563.80
PV ϭ $15,024.31
PV ϭ $7,121.66
6.77% semiannual
3.33% quarterly
1.10% monthly
PV1 ϭ $129,346.65
PV2 ϭ $124,854.21
G: 10.63%
H: 10.50%
114 payments
Balloon payment ϭ $348,430.68
PV ϭ $26,092,064.36
Profit ϭ $7,122.29
Break-even ϭ 18.56%
PV ϭ $3,356,644.06
PV ϭ $96,162.01
Value ϭ $5,614.47
PV5 ϭ $183,255.87
PV10 ϭ $281,961.41
PV ϭ $2,038.79; $2,252.86
Third year: $1,599.10
Life of loan: $7,740.97
EAR ϭ 12.36%
EAR ϭ 15.46%
Refundable fee:
EAR ϭ 8.92%
APR ϭ 8.57%
Nonrefundable fee:

EAR ϭ 8.84%
APR ϭ 8.50%

933

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934

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

6.70
6.72
6.74

a. $4,730.88
b. $48,603.46
c. $2,709.85
14.52%
PV ϭ $12,165.86
C ϭ $19,184.10

Chapter 7
7.4
7.6
7.8
7.10

7.12
7.18

7.24
7.26

7.28

8.76%
$1,076.43
7.13%
6.61%
8.65%
Current yield ϭ 9.62%
YTM ϭ 9.21%
Effective yield ϭ 9.42%
a. 10,000 coupon bonds; 132,679 zeroes
b. $10.9 million; $132.679 million
P: Current yield ϭ 8.97%
Capital gains yield ϭ Ϫ.97%
D: Current yield ϭ 6.78%
Capital gains yield ϭ 1.22%
PM ϭ $9,837.00
PN ϭ $1,944.44

Chapter 8
8.2
8.4
8.6
8.8

8.10
8.12
8.14
8.16
8.18
8.20
8.22

10.21%
$44.44
$3.93
6.85%
$26.91
$22.89
$2.98
$1.67
Close ϭ $35.97
Net income ϭ $1.44 million
a. $45.00
b. $47.30
10.25%

Chapter 9
9.4

9.6
9.8
9.10

Appendix C: Answers to

Selected End−of−Chapter
Problems

© The McGraw−Hill
Companies, 2002

APPENDIX C Answers to Selected End-of-Chapter Problems

C-2

6.66

Back Matter

a. 1.29 years
b. 2.14 years
c. 3.01 years
AAR ϭ 20.81%
@ 11%: NPV ϭ $4,658.40
@ 21%: NPV ϭ Ϫ$247.76
IRR ϭ 25.43%

9.12 a. IRRA ϭ 15.86%
IRRB ϭ 14.69%
b. NPVA ϭ $1,520.71
NPVB ϭ $1,698.58
c. Crossover rate ϭ 12.18%
9.14 a. @ 10%: NPV ϭ $13,570,247.93
b. IRR ϭ ϩ72.75%,
Ϫ83.46%

9.16 a. PII ϭ 1.266
PIII ϭ 2.109
b. NPVI ϭ $5,312.95
NPVII ϭ $3,328.24
9.18 @ 0%: NPV ϭ $128,252
@ ϱ%: NPV ϭ Ϫ$412,670
@ 14.57%: NPV ϭ 0
9.20 a. C ϭ I/N
b. C Ͼ I/PVIFAR%,N
c. C ϭ 2.0*I/PVIFAR%,N
9.22 IRR ϭ 25%, 33.33%, 42.86%, 66.67%

Chapter 10
10.2
10.4
10.8
10.10
10.12

10.14
10.16

10.18
10.20
10.22
10.26

Annual sales ϭ $339 million
OCF ϭ $277,561
Tax shield ϭ $38,080

Salvage value ϭ $1,548,032
OCF ϭ $927,500
CF0 ϭ Ϫ$2,375,000
CF1 ϭ $927,500
CF2 ϭ $927,500
CF3 ϭ $1,413,750
NPV ϭ $62,408.56
NPV ϭ $6,408.24
NPV ϭ $85,839.44
NPV ϭ Ϫ$108,550.35
Break-even cost savings ϭ $255,841.59
EACI ϭ Ϫ$78,263.13
EACII ϭ Ϫ$75,661.96
NPV ϭ Ϫ$26,574.44
EACA ϭ Ϫ$215,663.74
EACB ϭ Ϫ$159,470.87
Annual cost savings ϭ $163,515.59

Chapter 11
11.2

11.6

Total costs ϭ $6,811,600
Marginal cost ϭ $42.94
Average cost ϭ $48.65
Minimum revenue ϭ $429,400
Best-case NPV ϭ $4,649,729
Worst-case NPV ϭ Ϫ$92,984

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix C: Answers to
Selected End−of−Chapter
Problems

APPENDIX C Answers to Selected End-of-Chapter Problems

11.8

11.10
11.12
11.14

11.18
11.20

11.22
11.26

11.28

D ϭ $828,200
P ϭ $80.36
VC ϭ $55.35
QF ϭ 21,596
OCF ϭ $112,500
DOL ϭ 2.33
FC ϭ $22,500
OCF9,000 ϭ $5,850
OCF11,000 ϭ $12,150
DOL ϭ 1.3654
DOLA ϭ 2.3214
Payback ϭ 2.996 yrs
NPV ϭ $7,388,052
IRR ϭ 27.59%
⌬NPV/⌬P ϭ $128,649
⌬NPV/⌬Q ϭ $926.27
⌬OCF/⌬Q ϭ ϩ$18.60
⌬NPV/⌬Q ϭ ϩ$65.42
Qmin ϭ 27,373
DOL ϭ 1.12435
⌬OCF ϭ ϩ3.212%

Rf ϭ 6.0%
Slope ϭ .0929
Rf ϭ 4.92%
C ϭ $365,625
R ϭ $184,375
13.26 ␤I ϭ 2.82
␴I ϭ 13.15%
␤II ϭ 0.60

␴II ϭ 23.53%

13.16
13.18
13.20
13.24

Chapter 14
14.4
14.6
14.8
14.10
14.12
14.14
14.16

Chapter 12
Rd ϭ ϩ2.42%; Rc ϭ Ϫ17.74%
a. $120
b. 11.11%
c. 6.84%
12.6 2.42%; 2.71%
12.10 a. 7.83%
b. 7.40%
12.14 1/6;
Ϫ13.1% to ϩ24.5%;
Ϫ22.5% to ϩ33.9%
12.18 a. .3227; .2483
b. .0287; .1112
c. .1190; .0228

12.2
12.4

14.20

Chapter 15
15.2
15.4
15.6
15.8

15.10
15.12

Chapter 13
E(RP) ϭ 16.42%
X: $7,000
Y: $3,000
13.6 E(RI ) ϭ 6.50%
13.8 E(RP) ϭ 15.70%
13.10 a. E(RP) ϭ 8.41%
b. ␴P2 ϭ .03029
␴P ϭ 17.41%
13.12 ␤i ϭ 2.2
13.14 ␤i ϭ 1.14

13.2
13.4

a. $12.50

b. $1.92
$94.10
a. D0 ϭ $912.82
b. E0 ϭ $419.55
Straight-bond value ϭ $907.99
Conversion value ϭ $1,100.00
Warrant price ϭ $2.27
a. NPVbase ϭ $134,958.72
b. Q Ͻ 4,175
a. $1,190,002
b. $1,032,501.21
$814.63

15.14
15.16

15.18
15.20
15.22

13.05%
20.66%
Pretax cost ϭ 9.03%
Aftertax cost ϭ 5.87%
Book value ϭ $90 million
Market value ϭ $63.7 million
Aftertax cost ϭ 5.01%
13.07%
a. E/V ϭ 0.2547
D/V ϭ 0.7453

b. E/V ϭ 0.7785
D/V ϭ 0.2215
a. 11.45%
b. 17.95%
a. D/V ϭ 0.2392
P/V ϭ 0.0880
E/V ϭ 0.6728
b. 13.47%
b. 9.50%
c. $6.629 million
Break-even cost ϭ $45,901,639
$3,565,917

935

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936

Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

APPENDIX C Answers to Selected End-of-Chapter Problems

C-4

Chapter 16
a. $60; anything Ͼ 0
b. 1.20 million; 4
c. $58.00; $2.00
16.4 $4,000; Ϫ$500
16.6 786,102
16.8 No change;
Declines by $1.67;
Declines by $4.17
16.12 $37.50
16.14 $7,500

16.2

Chapter 17
17.2
17.4

17.6

17.8
17.10
17.12

17.14
17.16

a. EPS ϭ $.62, $1.56, $2.03
b. EPS ϭ $.17, $1.73, $2.51
a. I: EPS ϭ $2.00

II: EPS ϭ $1.00
b. I: EPS ϭ $7.00
II: EPS ϭ $11.00
c. $300,000
a. EPS ϭ $8.88, $9.50, $8.00
b. EBIT ϭ $4,500
c. EBIT ϭ $4,500
d. EBIT ϭ $4,500
a. $700
b. $840
$5.6 million
a. 18.70%
b. 14.35%
c. 16.78%, 14.85%, 11.00%
$208,000, $225,500
V ϭ $140,833.33

Chapter 18
a. New shares issued ϭ 1,000
b. New shares issued ϭ 2,500
18.4 a. $42.00
b. $60.87
c. $49.12
d. $122.50
e. 166,667; 115,000; 142,500; 57,143
18.6 Shares outstanding ϭ 3,920
Price ϭ $37.50
18.8 Shares outstanding ϭ 378,000
Capital surplus ϭ $2,182,000
18.10 New borrowings ϭ $384

Capital outlays ϭ $864
18.12 a. $560,000
b. No dividend paid
18.14 P0 ϭ $23.21
D ϭ $14.27
18.2

© The McGraw−Hill
Companies, 2002

Appendix C: Answers to
Selected End−of−Chapter
Problems

18.18 a.
b.
c.
d.

D
.72D
.903D
Price drop ϭ 1.377D

Chapter 19
Cash ϭ $5,850
CA ϭ $9,100
19.4 a. I, I
b. I, N
c. D, D

d. D, D
e. D, N
f. I, I
19.6 Operating cycle ϭ 86.136 days
Cash cycle ϭ 36.784 days
19.8 a. $157.50; $195; $150; $155.25
b. $135; $157.50; $195; $150
c. $142.50; $170; $180; $151.75
19.10 a. $226,666.67
b. $74,285.71
c. $139,857.29
$144,442.86
$170,400.00
19.12 a. 6.697%
b. $329,254.41
19.16 a. 7.786%
b. 7.798%
19.2

Chapter 20
20.2

a. $120,000
Ϫ$100,000
$20,000
b. Ϫ$50,000
$70,000
20.4 a. $104,000
b. $3,466.67
c. $800

4.3 days
20.6 a. $24,000
b. 2.48 days
c. $24,000
d. $5.06
e. $14,500
20.8 a. $280,000
b. $73.12 per day
c. $2,232.76 per month
20.10 NPV ϭ $5.9 million
Net savings ϭ $236,000
20.12 114 customers per day

A ppendix 20A
20A.2

$1,224.74


Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate Edition

Back Matter

© The McGraw−Hill
Companies, 2002

Appendix C: Answers to
Selected End−of−Chapter

Problems

APPENDIX C Answers to Selected End-of-Chapter Problems

a. Opportunity cost ϭ $9.00
Trading cost ϭ $333.33
b. $1,825.74
20A.10 16.00%

20A.4

Chapter 24

Chapter 21

24.2
24.4
24.6
24.8

21.2
21.4
21.6
21.8
21.10
21.12

21.14
21.16
21.18

$17,260,274
$100,000
Sales ϭ $239,344
Accounts receivable turnover ϭ 5.984
$542,465.75
NPV ϭ $1,927,000
Carrying cost ϭ $3,825
Order cost ϭ $2,496
EOQ ϭ 137.33
Orders ϭ 64.37 per year
NPV ϭ $647,333.33
2,925
$323.93

24.10
24.12
24.14
24.16
24.18
24.20

24.22

A ppendix 21A
21A.2 a. 3/15, net 45
b. $210,000
d. NPV ϭ Ϫ$1,725,000
Break-even price ϭ $117.83
Break-even discount ϭ 10.89%

21A.4 b. $49.83
c. NPV ϭ Ϫ$592,877.72

Chapter 22
22.2

c. 10.65996 FF/£
.0938 £/FF
22.6 France: 3.82%
Japan: 1.77%
Switzerland: 3.24%
22.8 United States inflation 2.52% lower
22.10 b. DM 1.5731
22.12 b. Ϫ6.30%

Chapter 23
23.2

Loss ϭ $6,212.50
Profit ϭ $3,787.50

24.24

24.26
24.30

$7,408.18
$5.21
1.85%
Call ϭ $4.63

Put ϭ $2.26
Call delta ϭ 0.67
Put delta ϭ Ϫ0.33
You pay $253,110.14
$3.23
$6.81
Equity ϭ $2,549.18
Debt ϭ $8,450.82
Equity ϭ $4,583.30
Debt ϭ $15,416.70
Cost of debt ϭ 26.03%
a. $13,415,249.81
b. $8,584,750.19
c. 10.69%
d. $13,839,268.65
e. 10.60%
a. $37,040.91
b. $9,770.28
c. $27,270.63; 12.12%
d. $24,067.25; 14.62%
e. Bondholders lose $3,203.38
Stockholders gain $3,203.38
$11.14
1

Chapter 25
EPS ϭ $4.875
PE ϭ 16.15 times
25.10 Ratio ϭ .7670

25.8

Chapter 26
26.2
26.4
26.6
26.8

NAL ϭ Ϫ$5,810.78
$12,723.90
Ϫ$10,838.83
$1,202,431.92

937

C-5