Present Value and Future Value Tables
Table A-1 Future Value Interest Factors for One Dollar Compounded at k Percent for n Periods: FVIF k,n = (1 + k) n
?Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
20%
24%
25%
30%
1
1.0100
1.0200
1.0300
1.0400
1.0500
1.0600
1.0700
1.0800
1.0900
1.1000
1.1100
1.1200
1.1300
1.1400
1.1500
1.1600
1.2000
1.2400
1.2500
1.3000
2
1.0201
1.0404
1.0609
1.0816
1.1025
1.1236
1.1449
1.1664
1.1881
1.2100
1.2321
1.2544
1.2769
1.2996
1.3225
1.3456
1.4400
1.5376
1.5625
1.6900
3
1.0303
1.0612
1.0927
1.1249
1.1576
1.1910
1.2250
1.2597
1.2950
1.3310
1.3676
1.4049
1.4429
1.4815
1.5209
1.5609
1.7280
1.9066
1.9531
2.1970
4
1.0406
1.0824
1.1255
1.1699
1.2155
1.2625
1.3108
1.3605
1.4116
1.4641
1.5181
1.5735
1.6305
1.6890
1.7490
1.8106
2.0736
2.3642
2.4414
2.8561
5
1.0510
1.1041
1.1593
1.2167
1.2763
1.3382
1.4026
1.4693
1.5386
1.6105
1.6851
1.7623
1.8424
1.9254
2.0114
2.1003
2.4883
2.9316
3.0518
3.7129
6
1.0615
1.1262
1.1941
1.2653
1.3401
1.4185
1.5007
1.5869
1.6771
1.7716
1.8704
1.9738
2.0820
2.1950
2.3131
2.4364
2.9860
3.6352
3.8147
4.8268
7
1.0721
1.1487
1.2299
1.3159
1.4071
1.5036
1.6058
1.7138
1.8280
1.9487
2.0762
2.2107
2.3526
2.5023
2.6600
2.8262
3.5832
4.5077
4.7684
6.2749
8
1.0829
1.1717
1.2668
1.3686
1.4775
1.5938
1.7182
1.8509
1.9926
2.1436
2.3045
2.4760
2.6584
2.8526
3.0590
3.2784
4.2998
5.5895
5.9605
8.1573
9
1.0937
1.1951
1.3048
1.4233
1.5513
1.6895
1.8385
1.9990
2.1719
2.3579
2.5580
2.7731
3.0040
3.2519
3.5179
3.8030
5.1598
6.9310
7.4506
10.604
10
1.1046
1.2190
1.3439
1.4802
1.6289
1.7908
1.9672
2.1589
2.3674
2.5937
2.8394
3.1058
3.3946
3.7072
4.0456
4.4114
6.1917
8.5944
9.3132
13.786
11
1.1157
1.2434
1.3842
1.5395
1.7103
1.8983
2.1049
2.3316
2.5804
2.8531
3.1518
3.4785
3.8359
4.2262
4.6524
5.1173
7.4301
10.657
11.642
17.922
12
1.1268
1.2682
1.4258
1.6010
1.7959
2.0122
2.2522
2.5182
2.8127
3.1384
3.4985
3.8960
4.3345
4.8179
5.3503
5.9360
8.9161
13.215
14.552
23.298
13
1.1381
1.2936
1.4685
1.6651
1.8856
2.1329
2.4098
2.7196
3.0658
3.4523
3.8833
4.3635
4.8980
5.4924
6.1528
6.8858
10.699
16.386
18.190
30.288
14
1.1495
1.3195
1.5126
1.7317
1.9799
2.2609
2.5785
2.9372
3.3417
3.7975
4.3104
4.8871
5.5348
6.2613
7.0757
7.9875
12.839
20.319
22.737
39.374
15
1.1610
1.3459
1.5580
1.8009
2.0789
2.3966
2.7590
3.1722
3.6425
4.1772
4.7846
5.4736
6.2543
7.1379
8.1371
9.2655
15.407
25.196
28.422
51.186
16
1.1726
1.3728
1.6047
1.8730
2.1829
2.5404
2.9522
3.4259
3.9703
4.5950
5.3109
6.1304
7.0673
8.1372
9.3576
10.748
18.488
31.243
35.527
66.542
17
1.1843
1.4002
1.6528
1.9479
2.2920
2.6928
3.1588
3.7000
4.3276
5.0545
5.8951
6.8660
7.9861
9.2765
10.761
12.468
22.186
38.741
44.409
86.504
18
1.1961
1.4282
1.7024
2.0258
2.4066
2.8543
3.3799
3.9960
4.7171
5.5599
6.5436
7.6900
9.0243
10.575
12.375
14.463
26.623
48.039
55.511
112.455
19
1.2081
1.4568
1.7535
2.1068
2.5270
3.0256
3.6165
4.3157
5.1417
6.1159
7.2633
8.6128
10.197
12.056
14.232
16.777
31.948
59.568
69.389
146.192
20
1.2202
1.4859
1.8061
2.1911
2.6533
3.2071
3.8697
4.6610
5.6044
6.7275
8.0623
9.6463
11.523
13.743
16.367
19.461
38.338
73.864
86.736
190.050
21
1.2324
1.5157
1.8603
2.2788
2.7860
3.3996
4.1406
5.0338
6.1088
7.4002
8.9492
10.804
13.021
15.668
18.822
22.574
46.005
91.592
108.420
247.065
22
1.2447
1.5460
1.9161
2.3699
2.9253
3.6035
4.4304
5.4365
6.6586
8.1403
9.9336
12.100
14.714
17.861
21.645
26.186
55.206
113.574
135.525
321.184
23
1.2572
1.5769
1.9736
2.4647
3.0715
3.8197
4.7405
5.8715
7.2579
8.9543
11.026
13.552
16.627
20.362
24.891
30.376
66.247
140.831
169.407
417.539
24
1.2697
1.6084
2.0328
2.5633
3.2251
4.0489
5.0724
6.3412
7.9111
9.8497
12.239
15.179
18.788
23.212
28.625
35.236
79.497
174.631
211.758
542.801
25
1.2824
1.6406
2.0938
2.6658
3.3864
4.2919
5.4274
6.8485
8.6231
10.835
13.585
17.000
21.231
26.462
32.919
40.874
95.396
216.542
264.698
705.641
30
1.3478
1.8114
2.4273
3.2434
4.3219
5.7435
7.6123
10.063
13.268
17.449
22.892
29.960
39.116
50.950
66.212
85.850
237.376
634.820
807.794
*
35
1.4166
1.9999
2.8139
3.9461
5.5160
7.6861
10.677
14.785
20.414
28.102
38.575
52.800
72.069
98.100
133.176
180.314
590.668
*
*
*
36
1.4308
2.0399
2.8983
4.1039
5.7918
8.1473
11.424
15.968
22.251
30.913
42.818
59.136
81.437
111.834
153.152
209.164
708.802
*
*
*
40
1.4889
2.2080
3.2620
4.8010
7.0400
10.286
14.974
21.725
31.409
45.259
65.001
93.051
132.782
188.884
267.864
378.721
*
*
*
*
50
1.6446
2.6916
4.3839
7.1067
11.467
18.420
29.457
46.902
74.358
117.391
184.565
289.002
450.736
700.233
*
*
*
*
*
*
????????????????????????????????????????????????????????????Table A-2 Future Value Interest Factors for a One-Dollar Annuity Compouned at k Percent for n Periods: FVIFA k,n = [(1 + k) n – 1 ] / k
?Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
20%
24%
25%
30%
1
1.0000
1.0200
1.0300
1.0400
1.0500
1.0600
1.0700
1.0800
1.0900
1.1000
1.1100
1.1200
1.1300
1.1400
1.1500
1.1600
1.2000
1.2400
1.2500
1.3000
2
2.0100
2.0200
2.0300
2.0400
2.0500
2.0600
2.0700
2.0800
2.0900
2.1000
2.1100
2.1200
2.1300
2.1400
2.1500
2.1600
2.2000
2.2400
2.2500
2.3000
3
3.0301
3.0604
3.0909
3.1216
3.1525
3.1836
3.2149
3.2464
3.2781
3.3100
3.3421
3.3744
3.4069
3.4396
3.4725
3.5056
3.6400
3.7776
3.8125
3.9900
4
4.0604
4.1216
4.1836
4.2465
4.3101
4.3746
4.4399
4.5061
4.5731
4.6410
4.7097
4.7793
4.8498
4.9211
4.9934
5.0665
5.3680
5.6842
5.7656
6.1870
5
5.1010
5.2040
5.3091
5.4163
5.5256
5.6371
5.7507
5.8666
5.9847
6.1051
6.2278
6.3528
6.4803
6.6101
6.7424
6.8771
7.4416
8.0484
8.2070
9.0431
6
6.1520
6.3081
6.4684
6.6330
6.8019
6.9753
7.1533
7.3359
7.5233
7.7156
7.9129
8.1152
8.3227
8.5355
8.7537
8.9775
9.9299
10.980
11.259
12.756
7
7.2135
7.4343
7.6625
7.8983
8.1420
8.3938
8.6540
8.9228
9.2004
9.4872
9.7833
10.089
10.405
10.730
11.067
11.414
12.916
14.615
15.073
17.583
8
8.2857
8.5830
8.8923
9.2142
9.5491
9.8975
10.260
10.637
11.028
11.436
11.859
12.300
12.757
13.233
13.727
14.240
16.499
19.123
19.842
23.858
9
9.3685
9.7546
10.159
10.583
11.027
11.491
11.978
12.488
13.021
13.579
14.164
14.776
15.416
16.085
16.786
17.519
20.799
24.712
25.802
32.015
10
10.462
10.950
11.464
12.006
12.578
13.181
13.816
14.487
15.193
15.937
16.722
17.549
18.420
19.337
20.304
21.321
25.959
31.643
33.253
42.619
11
11.567
12.169
12.808
13.486
14.207
14.972
15.784
16.645
17.560
18.531
19.561
20.655
21.814
23.045
24.349
25.733
32.150
40.238
42.566
56.405
12
12.683
13.412
14.192
15.026
15.917
16.870
17.888
18.977
20.141
21.384
22.713
24.133
25.650
27.271
29.002
30.850
39.581
50.895
54.208
74.327
13
13.809
14.680
15.618
16.627
17.713
18.882
20.141
21.495
22.953
24.523
26.212
28.029
29.985
32.089
34.352
36.786
48.497
64.110
68.760
97.625
14
14.947
15.974
17.086
18.292
19.599
21.015
22.550
24.215
26.019
27.975
30.095
32.393
34.883
37.581
40.505
43.672
59.196
80.496
86.949
127.913
15
16.097
17.293
18.599
20.024
21.579
23.276
25.129
27.152
29.361
31.772
34.405
37.280
40.417
43.842
47.580
51.660
72.035
100.815
109.687
167.286
16
17.258
18.639
20.157
21.825
23.657
25.673
27.888
30.324
33.003
35.950
39.190
42.753
46.672
50.980
55.717
60.925
87.442
126.011
138.109
218.472
17
18.430
20.012
21.762
23.698
25.840
28.213
30.840
33.750
36.974
40.545
44.501
48.884
53.739
59.118
65.075
71.673
105.931
157.253
173.636
285.014
18
19.615
21.412
23.414
25.645
28.132
30.906
33.999
37.450
41.301
45.599
50.396
55.750
61.725
68.394
75.836
84.141
128.117
195.994
218.045
371.518
19
20.811
22.841
25.117
27.671
30.539
33.760
37.379
41.446
46.018
51.159
56.939
63.440
70.749
78.969
88.212
98.603
154.740
244.033
273.556
483.973
20
22.019
24.297
26.870
29.778
33.066
36.786
40.995
45.762
51.160
57.275
64.203
72.052
80.947
91.025
102.444
115.380
186.688
303.601
342.945
630.165
21
23.239
25.783
28.676
31.969
35.719
39.993
44.865
50.423
56.765
64.002
72.265
81.699
92.470
104.768
118.810
134.841
225.026
377.465
429.681
820.215
22
24.472
27.299
30.537
34.248
38.505
43.392
49.006
55.457
62.873
71.403
81.214
92.503
105.491
120.436
137.632
157.415
271.031
469.056
538.101
*
23
25.716
28.845
32.453
36.618
41.430
46.996
53.436
60.893
69.532
79.543
91.148
104.603
120.205
138.297
159.276
183.601
326.237
582.630
673.626
*
24
26.973
30.422
34.426
39.083
44.502
50.816
58.177
66.765
76.790
88.497
102.174
118.155
136.831
158.659
184.168
213.978
392.484
723.461
843.033
*
25
28.243
32.030
36.459
41.646
47.727
54.865
63.249
73.106
84.701
98.347
114.413
133.334
155.620
181.871
212.793
249.214
471.981
898.092
*
*
30
34.785
40.568
47.575
56.085
66.439
79.058
94.461
113.283
136.308
164.494
199.021
241.333
293.199
356.787
434.745
530.312
*
*
*
*
35
41.660
49.994
60.462
73.652
90.320
111.435
138.237
172.317
215.711
271.024
341.590
431.663
546.681
693.573
881.170
*
*
*
*
*
36
43.077
51.994
63.276
77.598
95.836
119.121
148.913
187.102
236.125
299.127
380.164
484.463
618.749
791.673
*
*
*
*
*
*
40
48.886
60.402
75.401
95.026
120.800
154.762
199.635
259.057
337.882
442.593
581.826
767.091
*
*
*
*
*
*
*
*
50
64.463
84.579
112.797
152.667
209.348
290.336
406.529
573.770
815.084
*
*
*
*
*
*
*
*
*
*
*
????????????????????????????????????????????????????????????
Present Value and Future Value Tables
Table A-3 Present Value Interest Factors for One Dollar Discounted at k Percent for n Periods: PVIF k,n = 1 / (1 + k) n
?Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
20%
24%
25%
30%
1
0.9901
0.9804
0.9709
0.9615
0.9524
0.9434
0.9346
0.9259
0.9174
0.9091
0.9009
0.8929
0.8850
0.8772
0.8696
0.8621
0.8333
0.8065
0.8000
0.7692
2
0.9803
0.9612
0.9426
0.9246
0.9070
0.8900
0.8734
0.8573
0.8417
0.8264
0.8116
0.7972
0.7831
0.7695
0.7561
0.7432
0.6944
0.6504
0.6400
0.5917
3
0.9706
0.9423
0.9151
0.8890
0.8638
0.8396
0.8163
0.7938
0.7722
0.7513
0.7312
0.7118
0.6931
0.6750
0.6575
0.6407
0.5787
0.5245
0.5120
0.4552
4
0.9610
0.9238
0.8885
0.8548
0.8227
0.7921
0.7629
0.7350
0.7084
0.6830
0.6587
0.6355
0.6133
0.5921
0.5718
0.5523
0.4823
0.4230
0.4096
0.3501
5
0.9515
0.9057
0.8626
0.8219
0.7835
0.7473
0.7130
0.6806
0.6499
0.6209
0.5935
0.5674
0.5428
0.5194
0.4972
0.4761
0.4019
0.3411
0.3277
0.2693
6
0.9420
0.8880
0.8375
0.7903
0.7462
0.7050
0.6663
0.6302
0.5963
0.5645
0.5346
0.5066
0.4803
0.4556
0.4323
0.4104
0.3349
0.2751
0.2621
0.2072
7
0.9327
0.8706
0.8131
0.7599
0.7107
0.6651
0.6227
0.5835
0.5470
0.5132
0.4817
0.4523
0.4251
0.3996
0.3759
0.3538
0.2791
0.2218
0.2097
0.1594
8
0.9235
0.8535
0.7894
0.7307
0.6768
0.6274
0.5820
0.5403
0.5019
0.4665
0.4339
0.4039
0.3762
0.3506
0.3269
0.3050
0.2326
0.1789
0.1678
0.1226
9
0.9143
0.8368
0.7664
0.7026
0.6446
0.5919
0.5439
0.5002
0.4604
0.4241
0.3909
0.3606
0.3329
0.3075
0.2843
0.2630
0.1938
0.1443
0.1342
0.0943
10
0.9053
0.8203
0.7441
0.6756
0.6139
0.5584
0.5083
0.4632
0.4224
0.3855
0.3522
0.3220
0.2946
0.2697
0.2472
0.2267
0.1615
0.1164
0.1074
0.0725
11
0.8963
0.8043
0.7224
0.6496
0.5847
0.5268
0.4751
0.4289
0.3875
0.3505
0.3173
0.2875
0.2607
0.2366
0.2149
0.1954
0.1346
0.0938
0.0859
0.0558
12
0.8874
0.7885
0.7014
0.6246
0.5568
0.4970
0.4440
0.3971
0.3555
0.3186
0.2858
0.2567
0.2307
0.2076
0.1869
0.1685
0.1122
0.0757
0.0687
0.0429
13
0.8787
0.7730
0.6810
0.6006
0.5303
0.4688
0.4150
0.3677
0.3262
0.2897
0.2575
0.2292
0.2042
0.1821
0.1625
0.1452
0.0935
0.0610
0.0550
0.0330
14
0.8700
0.7579
0.6611
0.5775
0.5051
0.4423
0.3878
0.3405
0.2992
0.2633
0.2320
0.2046
0.1807
0.1597
0.1413
0.1252
0.0779
0.0492
0.0440
0.0254
15
0.8613
0.7430
0.6419
0.5553
0.4810
0.4173
0.3624
0.3152
0.2745
0.2394
0.2090
0.1827
0.1599
0.1401
0.1229
0.1079
0.0649
0.0397
0.0352
0.0195
16
0.8528
0.7284
0.6232
0.5339
0.4581
0.3936
0.3387
0.2919
0.2519
0.2176
0.1883
0.1631
0.1415
0.1229
0.1069
0.0930
0.0541
0.0320
0.0281
0.0150
17
0.8444
0.7142
0.6050
0.5134
0.4363
0.3714
0.3166
0.2703
0.2311
0.1978
0.1696
0.1456
0.1252
0.1078
0.0929
0.0802
0.0451
0.0258
0.0225
0.0116
18
0.8360
0.7002
0.5874
0.4936
0.4155
0.3503
0.2959
0.2502
0.2120
0.1799
0.1528
0.1300
0.1108
0.0946
0.0808
0.0691
0.0376
0.0208
0.0180
0.0089
19
0.8277
0.6864
0.5703
0.4746
0.3957
0.3305
0.2765
0.2317
0.1945
0.1635
0.1377
0.1161
0.0981
0.0829
0.0703
0.0596
0.0313
0.0168
0.0144
0.0068
20
0.8195
0.6730
0.5537
0.4564
0.3769
0.3118
0.2584
0.2145
0.1784
0.1486
0.1240
0.1037
0.0868
0.0728
0.0611
0.0514
0.0261
0.0135
0.0115
0.0053
21
0.8114
0.6598
0.5375
0.4388
0.3589
0.2942
0.2415
0.1987
0.1637
0.1351
0.1117
0.0926
0.0768
0.0638
0.0531
0.0443
0.0217
0.0109
0.0092
0.0040
22
0.8034
0.6468
0.5219
0.4220
0.3418
0.2775
0.2257
0.1839
0.1502
0.1228
0.1007
0.0826
0.0680
0.0560
0.0462
0.0382
0.0181
0.0088
0.0074
0.0031
23
0.7954
0.6342
0.5067
0.4057
0.3256
0.2618
0.2109
0.1703
0.1378
0.1117
0.0907
0.0738
0.0601
0.0491
0.0402
0.0329
0.0151
0.0071
0.0059
0.0024
24
0.7876
0.6217
0.4919
0.3901
0.3101
0.2470
0.1971
0.1577
0.1264
0.1015
0.0817
0.0659
0.0532
0.0431
0.0349
0.0284
0.0126
0.0057
0.0047
0.0018
25
0.7798
0.6095
0.4776
0.3751
0.2953
0.2330
0.1842
0.1460
0.1160
0.0923
0.0736
0.0588
0.0471
0.0378
0.0304
0.0245
0.0105
0.0046
0.0038
0.0014
30
0.7419
0.5521
0.4120
0.3083
0.2314
0.1741
0.1314
0.0994
0.0754
0.0573
0.0437
0.0334
0.0256
0.0196
0.0151
0.0116
0.0042
0.0016
0.0012
*
35
0.7059
0.5000
0.3554
0.2534
0.1813
0.1301
0.0937
0.0676
0.0490
0.0356
0.0259
0.0189
0.0139
0.0102
0.0075
0.0055
0.0017
0.0005
*
*
36
0.6989
0.4902
0.3450
0.2437
0.1727
0.1227
0.0875
0.0626
0.0449
0.0323
0.0234
0.0169
0.0123
0.0089
0.0065
0.0048
0.0014
*
*
*
40
0.6717
0.4529
0.3066
0.2083
0.1420
0.0972
0.0668
0.0460
0.0318
0.0221
0.0154
0.0107
0.0075
0.0053
0.0037
0.0026
0.0007
*
*
*
50
0.6080
0.3715
0.2281
0.1407
0.0872
0.0543
0.0339
0.0213
0.0134
0.0085
0.0054
0.0035
0.0022
0.0014
0.0009
0.0006
*
*
*
*
????????????????????????????????????????????????????????????Table A-4 Present Value Interest Factors for a One-Dollar Annuity Discounted at k Percent for n Periods: PVIFA = [1 – 1/(1 + k)n] / k
?Period
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
20%
24%
25%
30%
1
0.9901
0.9804
0.9709
0.9615
0.9524
0.9434
0.9346
0.9259
0.9174
0.9091
0.9009
0.8929
0.8850
0.8772
0.8696
0.8621
0.8333
0.8065
0.8000
0.7692
2
1.9704
1.9416
1.9135
1.8861
1.8594
1.8334
1.8080
1.7833
1.7591
1.7355
1.7125
1.6901
1.6681
1.6467
1.6257
1.6052
1.5278
1.4568
1.4400
1.3609
3
2.9410
2.8839
2.8286
2.7751
2.7232
2.6730
2.6243
2.5771
2.5313
2.4869
2.4437
2.4018
2.3612
2.3216
2.2832
2.2459
2.1065
1.9813
1.9520
1.8161
4
3.9020
3.8077
3.7171
3.6299
3.5460
3.4651
3.3872
3.3121
3.2397
3.1699
3.1024
3.0373
2.9745
2.9137
2.8550
2.7982
2.5887
2.4043
2.3616
2.1662
5
4.8534
4.7135
4.5797
4.4518
4.3295
4.2124
4.1002
3.9927
3.8897
3.7908
3.6959
3.6048
3.5172
3.4331
3.3522
3.2743
2.9906
2.7454
2.6893
2.4356
6
5.7955
5.6014
5.4172
5.2421
5.0757
4.9173
4.7665
4.6229
4.4859
4.3553
4.2305
4.1114
3.9975
3.8887
3.7845
3.6847
3.3255
3.0205
2.9514
2.6427
7
6.7282
6.4720
6.2303
6.0021
5.7864
5.5824
5.3893
5.2064
5.0330
4.8684
4.7122
4.5638
4.4226
4.2883
4.1604
4.0386
3.6046
3.2423
3.1611
2.8021
8
7.6517
7.3255
7.0197
6.7327
6.4632
6.2098
5.9713
5.7466
5.5348
5.3349
5.1461
4.9676
4.7988
4.6389
4.4873
4.3436
3.8372
3.4212
3.3289
2.9247
9
8.5660
8.1622
7.7861
7.4353
7.1078
6.8017
6.5152
6.2469
5.9952
5.7590
5.5370
5.3282
5.1317
4.9464
4.7716
4.6065
4.0310
3.5655
3.4631
3.0190
10
9.4713
8.9826
8.5302
8.1109
7.7217
7.3601
7.0236
6.7101
6.4177
6.1446
5.8892
5.6502
5.4262
5.2161
5.0188
4.8332
4.1925
3.6819
3.5705
3.0915
11
10.368
9.7868
9.2526
8.7605
8.3064
7.8869
7.4987
7.1390
6.8052
6.4951
6.2065
5.9377
5.6869
5.4527
5.2337
5.0286
4.3271
3.7757
3.6564
3.1473
12
11.255
10.575
9.9540
9.3851
8.8633
8.3838
7.9427
7.5361
7.1607
6.8137
6.4924
6.1944
5.9176
5.6603
5.4206
5.1971
4.4392
3.8514
3.7251
3.1903
13
12.134
11.348
10.635
9.9856
9.3936
8.8527
8.3577
7.9038
7.4869
7.1034
6.7499
6.4235
6.1218
5.8424
5.5831
5.3423
4.5327
3.9124
3.7801
3.2233
14
13.004
12.106
11.296
10.563
9.8986
9.2950
8.7455
8.2442
7.7862
7.3667
6.9819
6.6282
6.3025
6.0021
5.7245
5.4675
4.6106
3.9616
3.8241
3.2487
15
13.865
12.849
11.938
11.118
10.380
9.7122
9.1079
8.5595
8.0607
7.6061
7.1909
6.8109
6.4624
6.1422
5.8474
5.5755
4.6755
4.0013
3.8593
3.2682
16
14.718
13.578
12.561
11.652
10.838
10.106
9.4466
8.8514
8.3126
7.8237
7.3792
6.9740
6.6039
6.2651
5.9542
5.6685
4.7296
4.0333
3.8874
3.2832
17
15.562
14.292
13.166
12.166
11.274
10.477
9.7632
9.1216
8.5436
8.0216
7.5488
7.1196
6.7291
6.3729
6.0472
5.7487
4.7746
4.0591
3.9099
3.2948
18
16.398
14.992
13.754
12.659
11.690
10.828
10.059
9.3719
8.7556
8.2014
7.7016
7.2497
6.8399
6.4674
6.1280
5.8178
4.8122
4.0799
3.9279
3.3037
19
17.226
15.678
14.324
13.134
12.085
11.158
10.336
9.6036
8.9501
8.3649
7.8393
7.3658
6.9380
6.5504
6.1982
5.8775
4.8435
4.0967
3.9424
3.3105
20
18.046
16.351
14.877
13.590
12.462
11.470
10.594
9.8181
9.1285
8.5136
7.9633
7.4694
7.0248
6.6231
6.2593
5.9288
4.8696
4.1103
3.9539
3.3158
21
18.857
17.011
15.415
14.029
12.821
11.764
10.836
10.017
9.2922
8.6487
8.0751
7.5620
7.1016
6.6870
6.3125
5.9731
4.8913
4.1212
3.9631
3.3198
22
19.660
17.658
15.937
14.451
13.163
12.042
11.061
10.201
9.4424
8.7715
8.1757
7.6446
7.1695
6.7429
6.3587
6.0113
4.9094
4.1300
3.9705
3.3230
23
20.456
18.292
16.444
14.857
13.489
12.303
11.272
10.371
9.5802
8.8832
8.2664
7.7184
7.2297
6.7921
6.3988
6.0442
4.9245
4.1371
3.9764
3.3254
24
21.243
18.914
16.936
15.247
13.799
12.550
11.469
10.529
9.7066
8.9847
8.3481
7.7843
7.2829
6.8351
6.4338
6.0726
4.9371
4.1428
3.9811
3.3272
25
22.023
19.523
17.413
15.622
14.094
12.783
11.654
10.675
9.8226
9.0770
8.4217
7.8431
7.3300
6.8729
6.4641
6.0971
4.9476
4.1474
3.9849
3.3286
30
25.808
22.396
19.600
17.292
15.372
13.765
12.409
11.258
10.274
9.4269
8.6938
8.0552
7.4957
7.0027
6.5660
6.1772
4.9789
4.1601
3.9950
3.3321
35
29.409
24.999
21.487
18.665
16.374
14.498
12.948
11.655
10.567
9.6442
8.8552
8.1755
7.5856
7.0700
6.6166
6.2153
4.9915
4.1644
3.9984
3.3330
36
30.108
25.489
21.832
18.908
16.547
14.621
13.035
11.717
10.612
9.6765
8.8786
8.1924
7.5979
7.0790
6.6231
6.2201
4.9929
4.1649
3.9987
3.3331
40
32.835
27.355
23.115
19.793
17.159
15.046
13.332
11.925
10.757
9.7791
8.9511
8.2438
7.6344
7.1050
6.6418
6.2335
4.9966
4.1659
3.9995
3.3332
50
39.196
31.424
25.730
21.482
18.256
15.762
13.801
12.233
10.962
9.9148
9.0417
8.3045
7.6752
7.1327
6.6605
6.2463
4.9995
4.1666
3.9999
3.3333
????????????????????????????????????????????????????????????

Net Present Value
Last Time
We spent the time developing our basic approach to DCF analysis.
We discussed:
The importance of a financial market to the economy and why investors receive interest (compensation) for saving/lending.
The usefulness of the price from this market for decision making concerning real investments.
Now we want to complicate things.

Valuing Streams of Structured Future Cash Flows
Now we are going to discuss the valuation of certain highly structured cash flow streams.
The resulting valuation formulas are useful for simplifying the analysis of certain situations.
Pay attention to the exact timing of the cash flows, the formulas don’ t work unless you get this right.
Drawing diagrams of the cash flows can be useful.
These formulas can make life easier and so are worth understanding.
Perpetuity
A stream of equal payments, starting in one period, and made each period, forever. Forever??

Please, please remember, this gives the value of this stream of cash flows as of time 0, one period before the first payment arrives.
Growing Perpetuity
A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever.

The present value of a perpetuity that pays the amount C1 next period, grows at the rate g indefinitely when the discount rate is r is:

Annuities
An annuity is a series of equal payments, starting next period, and made each period for a specified number (3) of periods.

If payments occur at the end of each period (the first is one period from now) it is an ordinary annuity or an annuity in arrears.
If the payments occur at the beginning of each period (the first occurs now) it is an annuity in advance or an annuity due.
Valuing Annuities
We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1.
The present value of a T period annuity paying a periodic cash flow of C, when the discount rate is r, is:

If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r).
Or we can add C to the value of a T-1 period annuity.
Annuity Example
Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%.

or,

Annuity Due Example
What if the last example had the payments at the beginning of each period not the end?

Or,

Or,

Growing Annuities
A stream of payments each period for a fixed number of periods where the payment grows each period at a constant rate.
Example
What is the present value of a 20 year annuity with the first payment equal to $500, where the payments grow by 2% each year, when the interest rate is 10%?

Application: Retirement Planning
You have determined that you will require $2.5 million when you retire 25 years from now. Assuming an interest rate of r = 7%, how much should you set aside each year from now till retirement?
Step 1: Determine the present equivalent of the targeted $2.5 million.
PV = $2,500,000/(1.07)25
PV = $2,500,000/5.42743 = $460,623
Step 2: Determine the annuity that has an equivalent present value:
Retirement Planning cont
Now suppose that you expect your income to grow at 4% and you want to let your retirement contributions grow with your earnings. How large will the first contribution be? How about the last?
A College Planning Example
You have determined that you will need $60,000 per year for four years to send your daughter to college. The first of the four payments will be made 18 years from now and the last will be made 21 years from now. You wish to fund this obligation by making equal annual deposits at the end of each of the next 21 years. You expect to earn 8% per year on the deposits.
Step 1: Determine the t = 17 value of the obligation.

Step 2: Determine the equivalent t = 0 amount.
College Planning cont
Step 3: Determine the 21-year annuity that is equivalent to the stipulated present value.

Present Value Homework Problem
Your child will enter college 5 years from now. Tuition is expected to be $15,000 per year for (hopefully) 4 years (t=5,6,7,8).
You plan to make equal yearly deposits into an account at the end of each of the next 4 years (t=1,2,3,4) to fund tuition. The interest rate is 7%.
How much must you deposit each year?
What if tuition were growing by 2% each year over the 4 years?
Think about: How to decide whether/when to refinance your house?

Net Present Value Analysis – Converting Wheat to Native Range

For investment decisions and decisions that have financial impacts spanning more than one year, a more appropriate tool for decision-making is one called net present value (NPV). This technique uses a discount factor to put income and expenses in future years on a current year basis. The important variables in the decision are the level and timing of the income and expenses, the length of time over which the investment is used, and the discount factor (typically, an opportunity cost for money such as an interest rate) . The present value of annual cash flows are summed. If the total is positive, it indicates the benefits of the investment are larger than the cost of funds used to make the investment; if the total is negative, it indicates that the income from the investment is less than the cost of funds to make the investment. Note that NPV does not say that this is the best use of resources; it simply indicates whether the financial impacts of the investment over time are positive relative to the financing costs. Nor does NPV indicate the financial feasibility of an investment, that is, whether loan payments could be made if money were borrowed to make the investment.
Here’s an example of a NPV calculation made for the High Noon Ranch for converting wheat to native range with forbs and legumes. The time frame is 10 years as that is the expected life of the stand, given proper management and appropriate stocking rates. The discount rate chosen was 7%.

Table 4. NPV of converting wheat to native range with forbs

Year
Column 1

Cash Income
Column 2

Cash Expenses
Column 3

Annual Net Cash Flow
(1-2)
Column 4

Discount Factor
Column 5
Present Value of Annual Net Cash Flow
(3×4)
0

89.93
-89.93
1
-89.93
1
0
0
0
0.9346
0
2
0
0
0
0.8734
0
3
23.18
14.47
8.71
0.8163
7.11
4
23.18
14.47
8.71
0.7629
6.64
5
23.18
14.47
8.71
0.7130
6.21
6
23.18
14.47
8.71
0.6663
5.80
7
23.18
14.47
8.71
0.6227
5.42
8
23.18
14.47
8.71
0.5820
5.07
9
23.18
14.47
8.71
0.5439
4.72
10
23.18
14.47
8.71
0.5083
4.43
Total
-44.51

Here, the NPV calculation indicates the investment should not be made since the sum of the discounted annual cash flows is negative. However, the answer could be different if fewer than 7 acres were needed per cow, the stand had an expected life longer than 10 years, income were higher, costs were lower, or a different interest rate was appropriate.
The discount factor is calculated as 1/[(1+I)n] where I is the interest rate and n is the number of annual periods over which the sum is to be compounded. It can be calculated or referenced in farm management or agricultural finance textbooks.
Establishment costs of native range (table 2, pg. 15).
Cash income and cash expenses from spring cow-calf enterprise (page 13) with income per head divided by 17 assuming 17 acres per cow. Thus, cash income per acre is $394.44/17 acres; cash expenses are $245.96/17 acres. No income is received until the range is established in Year 3.

Benefit-Cost Measures
Lecture Goals:
Three BCA tools:
Net Present Value (NPV)
NPV Formula
Key Point
NPV Example
Benefit Cost Ratio (BCR)
BCR Formula
Key Point
BCR Example
Internal Rate of Return (IRR)
IRR Formula
Key Point
IRR Example
Advantages of BCA
Disadvantages of BCA

Chapter 18 Real Estate Finance Tools: Present Value and Mortgage Mathematics
Major Topics
Introduction to the Time Value of Money
Present & Future Value of a Single Sum
PV & FV over Multiple Periods of Time (Contd.)
PV of an Annuity
PV of Annuity (Contd.)

Calculating a Loan Balance
Calculating the Principal and Interest Separation of a Mortgage (Contd.)
Future Value of an Annuity
Calculating Yields or Borrowing Costs
More Mortgage Calcs on a Financial Calculator
The payment is based on the annuity that equates to a present value of the mortgage loan when discounted at the contract rate of interest
Effective Yield Calculation
Effective Yield Calculation
Annual Percentage Rate (APR)
Points -A tool to increase Yield
Mortgage Pricing (Contd.)
ARM and FRM
ARM and FRM
Choosing b/w FRMs and ARMs
Refinancing
END