confirming pages
A
Present Value Tables
APPENDIX TABLE 1
Discount factors: Present value of $1 to be received after t years ⫽ 1/(1 ⫹ r )
t
.
Number
of Years
Interest Rate per Year
1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%
1 .990 .980 .971 .962 .952 .943 .935 .926 .917 .909 .901 .893 .885 .877 .870
2 .980 .961 .943 .925 .907 .890 .873 .857 .842 .826 .812 .797 .783 .769 .756
3 .971 .942 .915 .889 .864 .840 .816 .794 .772 .751 .731 .712 .693 .675 .658
4 .961 .924 .888 .855 .823 .792 .763 .735 .708 .683 .659 .636 .613 .592 .572
5 .951 .906 .863 .822 .784 .747 .713 .681 .650 .621 .593 .567 .543 .519 .497
6 .942 .888 .837 .790 .746 .705 .666 .630 .596 .564 .535 .507 .480 .456 .432
7 .933 .871 .813 .760 .711 .665 .623 .583 .547 .513 .482 .452 .425 .400 .376
8 .923 .853 .789 .731 .677 .627 .582 .540 .502 .467 .434 .404 .376 .351 .327
9 .914 .837 .766 .703 .645 .592 .544 .500 .460 .424 .391 .361 .333 .308 .284
10 .905 .820 .744 .676 .614 .558 .508 .463 .422 .386 .352 .322 .295 .270 .247
11 .896 .804 .722 .650 .585 .527 .475 .429 .388 .350 .317 .287 .261 .237 .215
12 .887 .788 .701 .625 .557 .497 .444 .397 .356 .319 .286 .257 .231 .208 .187
13 .879 .773 .681 .601 .530 .469 .415 .368 .326 .290 .258 .229 .204 .182 .163
14 .870 .758 .661 .577 .505 .442 .388 .340 .299 .263 .232 .205 .181 .160 .141
15 .861 .743 .642 .555 .481 .417 .362 .315 .275 .239 .209 .183 .160 .140 .123
16 .853 .728 .623 .534 .458 .394 .339 .292 .252 .218 .188 .163 .141 .123 .107
17 .844 .714 .605 .513 .436 .371 .317 .270 .231 .198 .170 .146 .125 .108 .093
18 .836 .700 .587 .494 .416 .350 .296 .250 .212 .180 .153 .130 .111 .095 .081
19 .828 .686 .570 .475 .396 .331 .277 .232 .194 .164 .138 .116 .098 .083 .070
20 .820 .673 .554 .456 .377 .312 .258 .215 .178 .149 .124 .104 .087 .073 .061
Number
of Years
Interest Rate per Year
16%17%18%19%20%21%22%23%24%25%26%27%28%29%30%
1 .862 .855 .847 .840 .833 .826 .820 .813 .806 .800 .794 .787 .781 .775 .769
2 .743 .731 .718 .706 .694 .683 .672 .661 .650 .640 .630 .620 .610 .601 .592
3 .641 .624 .609 .593 .579 .564 .551 .537 .524 .512 .500 .488 .477 .466 .455
4 .552 .534 .516 .499 .482 .467 .451 .437 .423 .410 .397 .384 .373 .361 .350
5 .476 .456 .437 .419 .402 .386 .370 .355 .341 .328 .315 .303 .291 .280 .269
6 .410 .390 .370 .352 .335 .319 .303 .289 .275 .262 .250 .238 .227 .217 .207
7 .354 .333 .314 .296 .279 .263 .249 .235 .222 .210 .198 .188 .178 .168 .159
8 .305 .285 .266 .249 .233 .218 .204 .191 .179 .168 .157 .148 .139 .130 .123
9 .263 .243 .225 .209 .194 .180 .167 .155 .144 .134 .125 .116 .108 .101 .094
10 .227 .208 .191 .176 .162 .149 .137 .126 .116 .107 .099 .092 .085 .078 .073
11 .195 .178 .162 .148 .135 .123 .112 .103 .094 .086 .079 .072 .066 .061 .056
12 .168 .152 .137 .124 .112 .102 .092 .083 .076 .069 .062 .057 .052 .047 .043
13 .145 .130 .116 .104 .093 .084 .075 .068 .061 .055 .050 .045 .040 .037 .033
14 .125 .111 .099 .088 .078 .069 .062 .055 .049 .044 .039 .035 .032 .028 .025
15 .108 .095 .084 .074 .065 .057 .051 .045 .040 .035 .031 .028 .025 .022 .020
16 .093 .081 .071 .062 .054 .047 .042 .036 .032 .028 .025 .022 .019 .017 .015
17 .080 .069 .060 .052 .045 .039 .034 .030 .026 .023 .020 .017 .015 .013 .012
18 .069 .059 .051 .044 .038 .032 .028 .024 .021 .018 .016 .014 .012 .010 .009
19 .060 .051 .043 .037 .031 .027 .023 .020 .017 .014 .012 .011 .009 .008 .007
20 .051 .043 .037 .031 .026 .022 .019 .016 .014 .012 .010 .008 .007 .006 .005
Note: For example, if the interest rate is 10% per year, the present value of $1 received at year 5 is $.621.
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Present Value Tables A-1
Number
of Years
Interest Rate per Year
1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%
1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 1.110 1.120 1.130 1.140 1.150
2 1.020 1.040 1.061 1.082 1.102 1.124 1.145 1.166 1.188 1.210 1.232 1.254 1.277 1.300 1.323
3 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331 1.368 1.405 1.443 1.482 1.521
4 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464 1.518 1.574 1.630 1.689 1.749
5 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611 1.685 1.762 1.842 1.925 2.011
6 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772 1.870 1.974 2.082 2.195 2.313
7 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949 2.076 2.211 2.353 2.502 2.660
8 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.851 1.993 2.144 2.305 2.476 2.658 2.853 3.059
9 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358 2.558 2.773 3.004 3.252 3.518
10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594 2.839 3.106 3.395 3.707 4.046
11 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853 3.152 3.479 3.836 4.226 4.652
12 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138 3.498 3.896 4.335 4.818 5.350
13 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452 3.883 4.363 4.898 5.492 6.153
14 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797 4.310 4.887 5.535 6.261 7.076
15 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177 4.785 5.474 6.254 7.138 8.137
16 1.173 1.373 1.605 1.873 2.183 2.540 2.952 3.426 3.970 4.595 5.311 6.130 7.067 8.137 9.358
17 1.184 1.400 1.653 1.948 2.292 2.693 3.159 3.700 4.328 5.054 5.895 6.866 7.986 9.276 10.76
18 1.196 1.428 1.702 2.026 2.407 2.854 3.380 3.996 4.717 5.560 6.544 7.690 9.024 10.58 12.38
19 1.208 1.457 1.754 2.107 2.527 3.026 3.617 4.316 5.142 6.116 7.263 8.613 10.20 12.06 14.23
20 1.220 1.486 1.806 2.191 2.653 3.207 3.870 4.661 5.604 6.727 8.062 9.646 11.52 13.74 16.37
APPENDIX TABLE 2
Future value of $1 after t years ⫽ (1 ⫹ r )
t
.
Number
of Years
Interest Rate per Year
16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%
1 1.160 1.170 1.180 1.190 1.200 1.210 1.220 1.230 1.240 1.250 1.260 1.270 1.280 1.290 1.300
2 1.346 1.369 1.392 1.416 1.440 1.464 1.488 1.513 1.538 1.563 1.588 1.613 1.638 1.664 1.690
3 1.561 1.602 1.643 1.685 1.728 1.772 1.816 1.861 1.907 1.953 2.000 2.048 2.097 2.147 2.197
4 1.811 1.874 1.939 2.005 2.074 2.144 2.215 2.289 2.364 2.441 2.520 2.601 2.684 2.769 2.856
5 2.100 2.192 2.288 2.386 2.488 2.594 2.703 2.815 2.932 3.052 3.176 3.304 3.436 3.572 3.713
6 2.436 2.565 2.700 2.840 2.986 3.138 3.297 3.463 3.635 3.815 4.002 4.196 4.398 4.608 4.827
7 2.826 3.001 3.185 3.379 3.583 3.797 4.023 4.259 4.508 4.768 5.042 5.329 5.629 5.945 6.275
8 3.278 3.511 3.759 4.021 4.300 4.595 4.908 5.239 5.590 5.960 6.353 6.768 7.206 7.669 8.157
9 3.803 4.108 4.435 4.785 5.160 5.560 5.987 6.444 6.931 7.451 8.005 8.595 9.223 9.893 10.60
10 4.411 4.807 5.234 5.695 6.192 6.728 7.305 7.926 8.594 9.313 10.09 10.92 11.81 12.76 13.79
11 5.117 5.624 6.176 6.777 7.430 8.140 8.912 9.749 10.66 11.64 12.71 13.86 15.11 16.46 17.92
12 5.936 6.580 7.288 8.064 8.916 9.850 10.87 11.99 13.21 14.55 16.01 17.61 19.34 21.24 23.30
13 6.886 7.699 8.599 9.596 10.70 11.92 13.26 14.75 16.39 18.19 20.18 22.36 24.76 27.39 30.29
14 7.988 9.007 10.15 11.42 12.84 14.42 16.18 18.14 20.32 22.74 25.42 28.40 31.69 35.34 39.37
15 9.266 10.54 11.97 13.59 15.41 17.45 19.74 22.31 25.20 28.42 32.03 36.06 40.56 45.59 51.19
16 10.75 12.33 14.13 16.17 18.49 21.11 24.09 27.45 31.24 35.53 40.36 45.80 51.92 58.81 66.54
17 12.47 14.43 16.67 19.24 22.19 25.55 29.38 33.76 38.74 44.41 50.85 58.17 66.46 75.86 86.50
18 14.46 16.88 19.67 22.90 26.62 30.91 35.85 41.52 48.04 55.51 64.07 73.87 85.07 97.86 112.5
19 16.78 19.75 23.21 27.25 31.95 37.40 43.74 51.07 59.57 69.39 80.73 93.81 108.9 126.2 146.2
20 19.46 23.11 27.39 32.43 38.34 45.26 53.36 62.82 73.86 86.74 101.7 119.1 139.4 162.9 190.0
Note: For example, if the interest rate is 10% per year, the investment of $1 today will be worth $1.611 at year 5.
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A-2 Present Value Tables
Number
of Years
Interest Rate per Year
1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%
1 .990 .980 .971 .962 .952 .943 .935 .926 .917 .909 .901 .893 .885 .877 .870
2 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736 1.713 1.690 1.668 1.647 1.626
3 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 2.487 2.444 2.402 2.361 2.322 2.283
4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170 3.102 3.037 2.974 2.914 2.855
5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.993 3.890 3.791 3.696 3.605 3.517 3.433 3.352
6 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355 4.231 4.111 3.998 3.889 3.784
7 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868 4.712 4.564 4.423 4.288 4.160
8 7.652 7.325 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335 5.146 4.968 4.799 4.639 4.487
9 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759 5.537 5.328 5.132 4.946 4.772
10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145 5.889 5.650 5.426 5.216 5.019
11 10.37 9.787 9.253 8.760 8.306 7.887 7.499 7.139 6.805 6.495 6.207 5.938 5.687 5.453 5.234
12 11.26 10.58 9.954 9.385 8.863 8.384 7.943 7.536 7.161 6.814 6.492 6.194 5.918 5.660 5.421
13 12.13 11.35 10.63 9.986 9.394 8.853 8.358 7.904 7.487 7.103 6.750 6.424 6.122 5.842 5.583
14 13.00 12.11 11.30 10.56 9.899 9.295 8.745 8.244 7.786 7.367 6.982 6.628 6.302 6.002 5.724
15 13.87 12.85 11.94 11.12 10.38 9.712 9.108 8.559 8.061 7.606 7.191 6.811 6.462 6.142 5.847
16 14.72 13.58 12.56 11.65 10.84 10.11 9.447 8.851 8.313 7.824 7.379 6.974 6.604 6.265 5.954
17 15.56 14.29 13.17 12.17 11.27 10.48 9.763 9.122 8.544 8.022 7.549 7.120 6.729 6.373 6.047
18 16.40 14.99 13.75 12.66 11.69 10.83 10.06 9.372 8.756 8.201 7.702 7.250 6.840 6.467 6.128
19 17.23 15.68 14.32 13.13 12.09 11.16 10.34 9.604 8.950 8.365 7.839 7.366 6.938 6.550 6.198
20 18.05 16.35 14.88 13.59 12.46 11.47 10.59 9.818 9.129 8.514 7.963 7.469 7.025 6.623 6.259
A P P E N D I X T A B L E 3
Annuity table: Present value of $1 per year for each of t years ⫽ 1/ r ⫺ 1/[ r (1 ⫹ r )
t
] .
Number
of Years
Interest Rate per Year
16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%
1 .862 .855 .847 .840 .833 .826 .820 .813 .806 .800 .794 .787 .781 .775 .769
2 1.605 1.585 1.566 1.547 1.528 1.509 1.492 1.474 1.457 1.440 1.424 1.407 1.392 1.376 1.361
3 2.246 2.210 2.174 2.140 2.106 2.074 2.042 2.011 1.981 1.952 1.923 1.896 1.868 1.842 1.816
4 2.798 2.743 2.690 2.639 2.589 2.540 2.494 2.448 2.404 2.362 2.320 2.280 2.241 2.203 2.166
5 3.274 3.199 3.127 3.058 2.991 2.926 2.864 2.803 2.745 2.689 2.635 2.583 2.532 2.483 2.436
6 3.685 3.589 3.498 3.410 3.326 3.245 3.167 3.092 3.020 2.951 2.885 2.821 2.759 2.700 2.643
7 4.039 3.922 3.812 3.706 3.605 3.508 3.416 3.327 3.242 3.161 3.083 3.009 2.937 2.868 2.802
8 4.344 4.207 4.078 3.954 3.837 3.726 3.619 3.518 3.421 3.329 3.241 3.156 3.076 2.999 2.925
9 4.607 4.451 4.303 4.163 4.031 3.905 3.786 3.673 3.566 3.463 3.366 3.273 3.184 3.100 3.019
10 4.833 4.659 4.494 4.339 4.192 4.054 3.923 3.799 3.682 3.571 3.465 3.364 3.269 3.178 3.092
11 5.029 4.836 4.656 4.486 4.327 4.177 4.035 3.902 3.776 3.656 3.543 3.437 3.335 3.239 3.147
12 5.197 4.988 4.793 4.611 4.439 4.278 4.127 3.985 3.851 3.725 3.606 3.493 3.387 3.286 3.190
13 5.342 5.118 4.910 4.715 4.533 4.362 4.203 4.053 3.912 3.780 3.656 3.538 3.427 3.322 3.223
14 5.468 5.229 5.008 4.802 4.611 4.432 4.265 4.108 3.962 3.824 3.695 3.573 3.459 3.351 3.249
15 5.575 5.324 5.092 4.876 4.675 4.489 4.315 4.153 4.001 3.859 3.726 3.601 3.483 3.373 3.268
16 5.668 5.405 5.162 4.938 4.730 4.536 4.357 4.189 4.033 3.887 3.751 3.623 3.503 3.390 3.283
17 5.749 5.475 5.222 4.990 4.775 4.576 4.391 4.219 4.059 3.910 3.771 3.640 3.518 3.403 3.295
18 5.818 5.534 5.273 5.033 4.812 4.608 4.419 4.243 4.080 3.928 3.786 3.654 3.529 3.413 3.304
19 5.877 5.584 5.316 5.070 4.843 4.635 4.442 4.263 4.097 3.942 3.799 3.664 3.539 3.421 3.311
20 5.929 5.628 5.353 5.101 4.870 4.657 4.460 4.279 4.110 3.954 3.808 3.673 3.546 3.427 3.316
Note: For example, if the interest rate is 10% per year, the investment of $1 received in each of the next 5 years is $3.791.
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Present Value Tables A-3
rt .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.00 1.000 1.010 1.020 1.030 1.041 1.051 1.062 1.073 1.083 1.094
.10 1.105 1.116 1.127 1.139 1.150 1.162 1.174 1.185 1.197 1.209
.20 1.221 1.234 1.246 1.259 1.271 1.284 1.297 1.310 1.323 1.336
.30 1.350 1.363 1.377 1.391 1.405 1.419 1.433 1.448 1.462 1.477
.40 1.492 1.507 1.522 1.537 1.553 1.568 1.584 1.600 1.616 1.632
.50 1.649 1.665 1.682 1.699 1.716 1.733 1.751 1.768 1.786 1.804
.60 1.822 1.840 1.859 1.878 1.896 1.916 1.935 1.954 1.974 1.994
.70 2.014 2.034 2.054 2.075 2.096 2.117 2.138 2.160 2.181 2.203
.80 2.226 2.248 2.271 2.293 2.316 2.340 2.363 2.387 2.411 2.435
.90 2.460 2.484 2.509 2.535 2.560 2.586 2.612 2.638 2.664 2.691
1.00 2.718 2.746 2.773 2.801 2.829 2.858 2.886 2.915 2.945 2.974
1.10 3.004 3.034 3.065 3.096 3.127 3.158 3.190 3.222 3.254 3.287
1.20 3.320 3.353 3.387 3.421 3.456 3.490 3.525 3.561 3.597 3.633
1.30 3.669 3.706 3.743 3.781 3.819 3.857 3.896 3.935 3.975 4.015
1.40 4.055 4.096 4.137 4.179 4.221 4.263 4.306 4.349 4.393 4.437
1.50 4.482 4.527 4.572 4.618 4.665 4.711 4.759 4.807 4.855 4.904
1.60 4.953 5.003 5.053 5.104 5.155 5.207 5.259 5.312 5.366 5.419
1.70 5.474 5.529 5.585 5.641 5.697 5.755 5.812 5.871 5.930 5.989
1.80 6.050 6.110 6.172 6.234 6.297 6.360 6.424 6.488 6.553 6.619
1.90 6.686 6.753 6.821 6.890 6.959 7.029 7.099 7.171 7.243 7.316
2.00 7.389 7.463 7.538 7.614 7.691 7.768 7.846 7.925 8.004 8.085
2.10 8.166 8.248 8.331 8.415 8.499 8.585 8.671 8.758 8.846 8.935
2.20 9.025 9.116 9.207 9.300 9.393 9.488 9.583 9.679 9.777 9.875
2.30 9.974 10.07 10.18 10.28 10.38 10.49 10.59 10.70 10.80 10.91
2.40 11.02 11.13 11.25 11.36 11.47 11.59 11.70 11.82 11.94 12.06
2.50 12.18 12.30 12.43 12.55 12.68 12.81 12.94 13.07 13.20 13.33
2.60 13.46 13.60 13.74 13.87 14.01 14.15 14.30 14.44 14.59 14.73
2.70 14.88 15.03 15.18 15.33 15.49 15.64 15.80 15.96 16.12 16.28
2.80 16.44 16.61 16.78 16.95 17.12 17.29 17.46 17.64 17.81 17.99
2.90 18.17 18.36 18.54 18.73 18.92 19.11 19.30 19.49 19.69 19.89
3.00 20.09 20.29 20.49 20.70 20.91 21.12 21.33 21.54 21.76 21.98
3.10 22.20 22.42 22.65 22.87 23.10 23.34 23.57 23.81 24.05 24.29
3.20 24.53 24.78 25.03 25.28 25.53 25.79 26.05 26.31 26.58 26.84
3.30 27.11 27.39 27.66 27.94 28.22 28.50 28.79 29.08 29.37 29.67
3.40 29.96 30.27 30.57 30.88 31.19 31.50 31.82 32.14 32.46 32.79
3.50 33.12 33.45 33.78 34.12 34.47 34.81 35.16 35.52 35.87 36.23
3.60 36.60 36.97 37.34 37.71 38.09 38.47 38.86 39.25 39.65 40.04
3.70 40.45 40.85 41.26 41.68 42.10 42.52 42.95 43.38 43.82 44.26
3.80 44.70 45.15 45.60 46.06 46.53 46.99 47.47 47.94 48.42 48.91
3.90 49.40 49.90 50.40 50.91 51.42 51.94 52.46 52.98 53.52 54.05
Note: For example, if the continuously compounded interest rate is 10% per year, the investment of $1 today will be worth $1.105 at year 1 and $1.221 at year 2.
A P P E N D I X T A B L E 4
Values of e
rt
. Future value of $1 invested at a continuously compounded rate r for t years.
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A-4 Present Value Tables
APPENDIX TABLE 5
Present value of $1 per year received in a continuous stream for each of t years (discounted at an annually compounded rate r ) ⫽ {1 ⫺ 1/(1 ⫹ r )
t
} /
{ln(1 ⫹ r )}.
Number
of Years
Interest Rate per Year
1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%
1 .995 .990 .985 .981 .976 .971 .967 .962 .958 .954 .950 .945 .941 .937 .933
2 1.980 1.961 1.942 1.924 1.906 1.888 1.871 1.854 1.837 1.821 1.805 1.790 1.774 1.759 1.745
3 2.956 2.913 2.871 2.830 2.791 2.752 2.715 2.679 2.644 2.609 2.576 2.543 2.512 2.481 2.450
4 3.922 3.846 3.773 3.702 3.634 3.568 3.504 3.443 3.383 3.326 3.270 3.216 3.164 3.113 3.064
5 4.878 4.760 4.648 4.540 4.437 4.337 4.242 4.150 4.062 3.977 3.896 3.817 3.741 3.668 3.598
6 5.825 5.657 5.498 5.346 5.202 5.063 4.931 4.805 4.685 4.570 4.459 4.353 4.252 4.155 4.062
7 6.762 6.536 6.323 6.121 5.930 5.748 5.576 5.412 5.256 5.108 4.967 4.832 4.704 4.582 4.465
8 7.690 7.398 7.124 6.867 6.623 6.394 6.178 5.974 5.780 5.597 5.424 5.260 5.104 4.956 4.816
9 8.609 8.243 7.902 7.583 7.284 7.004 6.741 6.494 6.261 6.042 5.836 5.642 5.458 5.285 5.121
10 9.519 9.072 8.657 8.272 7.913 7.579 7.267 6.975 6.702 6.447 6.208 5.983 5.772 5.573 5.386
11 10.42 9.884 9.391 8.935 8.512 8.121 7.758 7.421 7.107 6.815 6.542 6.287 6.049 5.826 5.617
12 11.31 10.68 10.10 9.572 9.083 8.633 8.218 7.834 7.478 7.149 6.843 6.559 6.294 6.048 5.818
13 12.19 11.46 10.79 10.18 9.627 9.116 8.647 8.216 7.819 7.453 7.115 6.802 6.512 6.242 5.992
14 13.07 12.23 11.46 10.77 10.14 9.571 9.048 8.570 8.131 7.729 7.359 7.018 6.704 6.413 6.144
15 13.93 12.98 12.12 11.34 10.64 10.00 9.423 8.897 8.418 7.980 7.579 7.212 6.874 6.563 6.276
16 14.79 13.71 12.75 11.88 11.11 10.41 9.774 9.201 8.681 8.209 7.778 7.385 7.024 6.694 6.390
17 15.64 14.43 13.36 12.41 11.55 10.79 10.10 9.482 8.923 8.416 7.957 7.539 7.158 6.809 6.490
18 16.48 15.14 13.96 12.91 11.98 11.15 10.41 9.742 9.144 8.605 8.118 7.676 7.275 6.910 6.577
19 17.31 15.83 14.54 13.39 12.39 11.49 10.69 9.983 9.347 8.777 8.263 7.799 7.380 6.999 6.652
20 18.14 16.51 15.10 13.86 12.77 11.81 10.96 10.21 9.533 8.932 8.394 7.909 7.472 7.077 6.718
Note: For example, if the interest rate is 10% per year, a continuous cash flow of $1 a year for each of 5 years is worth $3.977. A continuous flow of $1 in year 5 only
is worth $3.977 ⫺ $3.326 ⫽ $.651.
Number
of Years
Interest Rate per Year
16% 17% 18% 19% 20% 21% 22% 23% 24% 25% 26% 27% 28% 29% 30%
1 .929 .925 .922 .918 .914 .910 .907 .903 .900 .896 .893 .889 .886 .883 .880
2 1.730 1.716 1.703 1.689 1.676 1.663 1.650 1.638 1.625 1.613 1.601 1.590 1.578 1.567 1.556
3 2.421 2.392 2.365 2.337 2.311 2.285 2.259 2.235 2.211 2.187 2.164 2.141 2.119 2.098 2.077
4 3.016 2.970 2.925 2.882 2.840 2.799 2.759 2.720 2.682 2.646 2.610 2.576 2.542 2.509 2.477
5 3.530 3.464 3.401 3.340 3.281 3.223 3.168 3.115 3.063 3.013 2.964 2.917 2.872 2.828 2.785
6 3.972 3.886 3.804 3.724 3.648 3.574 3.504 3.436 3.370 3.307 3.246 3.187 3.130 3.075 3.022
7 4.354 4.247 4.145 4.048 3.954 3.865 3.779 3.696 3.617 3.542 3.469 3.399 3.331 3.266 3.204
8 4.682 4.555 4.434 4.319 4.209 4.104 4.004 3.909 3.817 3.730 3.646 3.566 3.489 3.415 3.344
9 4.966 4.819 4.680 4.547 4.422 4.302 4.189 4.081 3.978 3.880 3.786 3.697 3.612 3.530 3.452
10 5.210 5.044 4.887 4.739 4.599 4.466 4.340 4.221 4.108 4.000 3.898 3.801 3.708 3.619 3.535
11 5.421 5.237 5.063 4.900 4.747 4.602 4.465 4.335 4.213 4.096 3.986 3.882 3.783 3.689 3.599
12 5.603 5.401 5.213 5.036 4.870 4.713 4.566 4.428 4.297 4.173 4.057 3.946 3.841 3.742 3.648
13 5.759 5.542 5.339 5.150 4.972 4.806 4.650 4.503 4.365 4.235 4.112 3.997 3.887 3.784 3.686
14 5.894 5.662 5.446 5.245 5.058 4.882 4.718 4.564 4.420 4.284 4.157 4.036 3.923 3.816 3.715
15 6.010 5.765 5.537 5.326 5.129 4.945 4.774 4.614 4.464 4.324 4.192 4.068 3.951 3.841 3.737
16 6.111 5.853 5.614 5.393 5.188 4.998 4.820 4.655 4.500 4.355 4.220 4.092 3.973 3.860 3.754
17 6.197 5.928 5.679 5.450 5.238 5.041 4.858 4.687 4.529 4.381 4.242 4.112 3.990 3.875 3.767
18 6.272 5.992 5.735 5.498 5.279 5.076 4.889 4.714 4.552 4.401 4.259 4.127 4.003 3.887 3.778
19 6.336 6.047 5.781 5.538 5.313 5.106 4.914 4.736 4.571 4.417 4.273 4.139 4.014 3.896 3.785
20 6.391 6.094 5.821 5.571 5.342 5.130 4.935 4.754 4.586 4.430 4.284 4.149 4.022 3.903 3.791
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confirming pages
Present Value Tables A-5
APPENDIX TABLE 6
Cumulative probability [ N ( d )] that a normally distributed variable will be less than d standard deviations above the mean.
Note: For example, if d ⫽ .22, N ( d ) ⫽ .5871 (i.e., there is a .5871 probability that a normally distributed variable will be less than .22 standard deviations above the mean).
d 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
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