Justus and Julius are twins. Beginning at age 25, Justus invests $3,000 per year for 10 years and then never sets aside another penny. Julius waits ten years (age 35) and then invests $3,000 per year for the next 30 years. Assuming they both earn 6%, how much each twin have at 65? (Hint: Use Appendixes A.1 and A.3)

Appendix A-1 Future Value of a Single Amount ($1 at the End Future Value of a Known Lump Sum) n 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 1 1.0100 1.0200 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000 1.1100 1.1200 1.0300 1.0609 2 1.0201 1.0404 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100 1.2321 1.2544 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 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 5 1.0510 1.1041 1.1593 1.2167 1.3382 1.4026 1.5386 1.6105 1.6851 1.7623 1.2763 1.3401 1.4693 1.5869 6 1.0615 1.1262 1.1941 1.2653 1.4185 1.5007 1.6771 1.7716 1.8704 1.9738 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 8 1.0829 1.1717 1.2668 1.3686 1,4775 1.7182 1.8509 1.9926 2.1436 2.3045 2.4760 1.5938 1.6895 9 1.0937 1.1951 1.3048 1.4233 1.5513 1.8385 1.9990 2.1719 2.3579 2.5580 2.7731 10 1.1046 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937 3.1058 2.8394 3.1518 11 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.8531 3.4785 2.5804 2.8127 12 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 3.1384 3.4985 3.8960 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 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 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 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 17 1.1843 1.4002 1.6528 1.9479 2.2920 2.6928 3.1588 3.7000 43276 5.0545 5.8951 6.8660 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 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 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 21 1.2324 1.5157 1.8603 2.2788 2.7860 3.3996 4.1406 5.0338 6.1088 7.4002 8.9492 10.8038 22 1.2447 1.5460 1.9161 2.3699 2.9253 3.6035 4.4304 5.4365 6.6586 8.1403 9.9336 12. 1003 23 1.2572 1.5769 1.9736 2.4647 3.0715 3.8197 4.7405 5.8715 7.2579 8.9543 11.0263 13.5523 24 1.2697 1.6084 2.0328 2.5633 3.2251 4,0489 5.0724 6.3412 7.9111 9.8497 12.2392 15.1786 25 1.2824 1.6406 2.6658 4.2919 5.4274 6.8485 10.8347 2.0938 2. 1566 3.3864 3.5557 8.6231 9.3992 17.0001 19.0401 26 1.2953 1.6734 2.7725 4.5494 5.8074 73964 13.5855 15.0799 16.7386 11.9182 27 1.3082 1.7069 2.2213 2.8834 3.7335 4.8223 6.2139 7.9881 10.2451 13.1100 21.3249 28 1.3213 1.7410 2.2879 2.9987 3.9201 5.1117 6.6488 8.6271 11.1671 14.4210 18.5799 23.8839 29 1.3345 1.7758 2.3566 3.1187 4.1161 5.4184 7.1143 9.3173 12.1722 15.8631 20.6237 26.7499 30 1.3478 1.8114 2.4273 3.2434 4.3219 5.7435 7.6123 10.0627 13.2677 17.4494 22.8923 29.9599 40 1.4889 2.2080 3.2620 4.8010 7.0400 10.2857 14.9745 21.7245 31.4094 45.2593 65.0009 93.0510 289.0022 50 1.6446 2.6916 4.3839 7.1067 11.4674 18.4202 29.4570 46.9016 74.3575 117.3909 184.5648 of n Periods) (Used to compute the Compounded 13% 14% 15% 16% 17% 18% 19% 20% 1.1300 1.1400 1.1500 1.1600 1.1700 1.1800 1.1900 1.2000 1.2769 1.2996 1.3225 1.3456 1.3689 1.3924 1.4161 1.4400 1.4429 1.4815 1.5209 1.5609 1.6016 1.6430 1.6852 1.7280 1.6305 1.6890 1.7490 1.8106 1.8739 1.9388 2.0736 2.0053 2.3864 1,8424 1.9254 2.0114 2. 1003 2.1924 2.2878 2.4883 2.0820 2.1950 2.3131 2.4364 2.5652 2.6996 2.8398 2.9860 2.3526 2.5023 2.6600 2.8262 3.0012 3.1855 3.3793 3.5832 2.6584 2.8526 3.0590 3.2784 3.5115 3.7589 4.0214 4.2998 3.2519 3.8030 4.4355 4.7854 5.1598 3.0040 3.3946 3.5179 4.0456 4.1084 4.8068 3.7072 4.4114 5.2338 5.6947 6.1917 3.8359 4.2262 4.6524 5.1173 5.6240 6.1759 6.7767 7.4301 4.3345 4.8179 5.3503 5.9360 6.5801 7.2876 8.0642 8.9161 4.8980 5.4924 6.1528 6.8858 7.6987 8.5994 9.5964 10.6993 5.5348 6.2613 7.0757 7.9875 9.0075 10.1472 11.4198 12.8392 6.2543 7.1379 8.1371 9.2655 10.5387 11.9737 13.5895 15.4070 7.0673 8.1372 9.3576 10.7480 12.3303 14.1290 16.1715 18.4884 7.9861 9.2765 12.4677 14.4265 16.6722 19.2441 22.1861 10.7613 12.3755 9.0243 10.5752 14.4625 16.8790 19.6733 22.9005 26.6233 10.1974 12.0557 14.2318 16.7765 19.7484 23.2144 27.2516 31.9480 11.5231 13.7435 16.3665 19.4608 23.1056 27.3930 32.4294 38.3376 15.6676 18.8215 22.5745 27.0336 32.3238 38.5910 46.0051 13.0211 14.7138 17.8610 21.6447 26.1864 31.6293 38.1421 45.9233 55.2061 16.6266 24.8915 37.0062 45.0076 54.6487 66.2474 20.3616 23.2122 30.3762 35.2364 18.7881 28.6252 43.2973 53.1090 65.0320 79.4968 21.2305 26.4619 32.9190 40.8742 50.6578 62.6686 95.3962 77.3881 92.0918 23.9905 30.1666 37.8568 47.4141 73.9490 114.4755 59.2697 69.3455 27.1093 34.3899 43.5353 55.0004 109.5893 137.3706 87.2598 102.9666 30.6335 39.2045 50.0656 63.8004 81.1342 130.4112 164.8447 34.6158 44.6931 57.5755 74.0085 94.9271 121.5005 155.1893 197.8136 39.1159 50.9502 66.2118 85.8499 111.0647 143.3706 184.6753 237.3763 267.8635 378.7212 132.7816 188.8835 450.7359 700.2330 533.8687 2566.215 750.3783 3927357 1051.668 1469.772 5988.914 9100.438 1083.657 1670.704 Appendix A-3 Future Value of a Series of Equal Amounts (an Annuity (Used to compute the Compounded Future Value of a n 1% 2% 6% 8% 1 1.0000 12% 1.0000 2.1200 3.3744 2 1.0000 2.0200 3.0604 4.1216 2.0100 3.0301 3 4 4.7793 5 4.0604 5.1010 6.1520 5.2040 6.3081 7.4343 6.3528 8.1152 6 7 7 7.2135 10.0890 8 8.2857 8.5830 12.2997 9 14.7757 9.3685 9.7546 10.4622 10.9497 10 17.5487 11 20.6546 11.5668 12.1687 12.6825 13.4121 12 13 14 13.8093 14.6803 14.9474 15.9739 16.0969 17.2934 17.2579 18.6393 15 3% 4% 5% 7% 9% 10% 11% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 2.1100 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 3.2781 3.3100 3.3421 4.1836 4.2465 4.3101 43746 4.4399 4.5061 4.5731 4.6410 4.7097 5.3091 5,4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6.2278 6,4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7.9129 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 9.7833 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 11.8594 10.1591 10.5828 11.0266 11.4913 11.9780 12.4876 13.0210 13.5795 14.1640 11,4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374 16.7220 12.8078 13.4864 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312 19.5614 14.1920 15.0258 15.9171 16.8699 17.8885 18.9771 20.1407 21.3843 22.7132 15.6178 16.6268 17.7130 18.8821 20.1406 21.4953 22.9534 24.5227 26.2116 17.0863 18.2919 19.5986 21.0151 22.5505 24.2149 26.0192 27.9750 30.0949 18.5989 20.0236 21.5786 23.2760 25.1290 27.1521 29.3609 31.7725 34.4054 20.1569 21.8245 23.6575 25.6725 27.8881 30.3243 33.0034 35.9497 39.1899 21.7616 23.6975 25.8404 28.2129 30.8402 33.7502 36.9737 40.5447 44.5008 23.4144 25.6454 28.1324 30.9057 33.9990 37.4502 41.3013 45.5992 50.3959 25.1169 27.6712 30.5390 33.7600 37.3790 41.4463 46.0185 51.1591 56.9395 26.8704 29.7781 33,0660 36.7856 40.9955 45.7620 51.1601 57.2750 64.2028 28.6765 31.9692 35.7193 39.9927 44.8652 50.4229 56.7645 64.0025 72.2651 30.5368 34.2480 38.5052 43.3923 49.0057 55.4568 62.8733 71.4027 81.2143 32.4529 36.6179 41.4305 46.9958 53.4361 60.8933 69.5319 79.5430 91.1479 34.4265 39.0826 44.5020 50.8156 58.1767 66.7648 76.7898 88.4973 102.1741 36.4593 41.6459 47.7271 54.8645 63.2490 73.1059 84.7009 98.3471 114.4133 38.5530 44.3117 51.1135 59.1564 68.6765 79.9544 93.3240 109.1818 127.9988 40.7096 47.0842 54.6691 63.7058 74.4838 87.3508 102.7231 121.0999 143.0786 42.9309 49.9676 58.4026 68.5281 80.6977 95.3388 112.9682 134.2099 159.8173 45.2188 52.9663 62.3227 73.6398 87 3465 103.9659 124.1354 148.6309 178.3972 47.5754 56.0849 66.4389 79.0582 94.4608 113.2832 136.3075 164.4940 199.0209 75.4013 95.0255 120.7998 154.7620 199.6351 259.0565 337.8824 442.5925 581.8260 112.7969 152.6671 209.3480 290.3359 406.5289 573.7701 815.0834 1163.908 1668.771 24.1331 28.0291 32.3926 37.2797 42.7533 48.8837 55.7497 63.4397 16 1 17 18 19 20 21 22 23 18.4304 20.0121 19.6147 21.4123 20.8109 22.8406 22.0190 24.2974 23.2392 25.7833 24.4716 27.2990 25.7163 28.8450 26.9735 30.4219 28.2432 32.0303 29.5256 33.6709 30.8209 35.3443 32.1291 37.0512 33.4504 38.7922 34.7849 40.5681 24 25 26 72.0524 81.6987 92.5026 104.6029 118.1552 133.3339 150.3339 169.3740 190.6989 214.5827 241.3327 767.0914 2400.018 27 28 29 30 40 48.8864 60.4020 64.4632 84.5794 50 of $1 Paid at the End of Each Period) Stream of Income Payments) 13% 14% 15% 16% 17% 18% 19% 20% 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.1300 2.1400 2.1500 2.1600 2.1700 2.1800 2.1900 2.2000 3.4069 3.4396 3.4725 3.5056 3.5389 3.5724 3.6061 3,6400 4.8498 4.9211 4.9934 5.0665 5.1405 5.2154 5.2913 5.3680 6.4803 6.6101 6.7424 6.8771 7.0144 7.2966 7.4416 7.1542 9.4420 8.7537 8.9775 9.2068 9.6830 9.9299 8.3227 10.4047 8.5355 10.7305 11.0668 11.4139 11.7720 12.1415 12.5227 12.9159 12.7573 13.2328 13.7268 14.2401 14.7733 15.3270 15.9020 16.4991 15.4157 16.0853 16.7858 17.5185 18.2847 19.0859 19.9234 20.7989 18.4197 19.3373 20.3037 21.3215 22.3931 23.5213 24.7089 25.9587 21.8143 243493 25.7329 27.1999 28.7551 30.4035 32.1504 23.0445 27.2707 25.6502 29.0017 30.8502 32.8239 34.9311 37.1802 39.5805 29.9847 32.0887 36.7862 39.4040 42.2187 45.2445 48.4966 34.3519 40.5047 34.8827 37.5811 43.6720 47.1027 50.8180 54.8409 59.1959 40.4175 43.8424 47.5804 51.6595 56.1101 60.9653 66.2607 72.0351 46.67 17 50.9804 55.7175 60.9250 66.6488 72.9390 87.4421 79.8502 96.0217 53.7391 59.1176 65.0751 71.6730 78.9791 105.9306 87.0680 103.7403 61.7251 68.3941 75.8364 84.1407 93.4056 115.2659 128.1167 70.7494 78.9692 88.2118 98.6032 110.2846 123.4135 138.1664 154.7400 80.9468 91.0249 115.3797 130.0329 146.6280 165.4180 186.6880 92.4699 104.7684 102.4436 118.8101 137.6316 153.1385 197.8474 134.8405 157.4150 174.0210 206.3448 225.0256 271.0307 105.4910 120.4360 180.1721 236.4384 282.3618 120.2048 138.2970 159.2764 183.6014 211.8013 244,4868 326,2368 136.8315 158.6586 184.1678 213.9776 248.8075 289.4945 3370105 392.4842 155.6196 249.2140 292.1048 402.0424 471.9811 181.8708 208.3327 212.7930 245.7120 342.6035 405.2721 176.8501 290.0883 342.7626 479.4305 567.3773 200.8406 238.4993 283.5688 337.5024 402.0323 479.2211 571.5223 681.8527 227.9499 272.8892 327.1041 392.5027 471.3778 566.4808 681.1116 819.2233 258.5834 312.0937 456.3032 552.5120 669.4474 984.0679 377.1697 434.7451 811.5228 966.7121 293.1992 356.7868 530.3117 647.4390 790.9479 1181.882 1013.704 1342.025 1779.090 2360.757 3134.522 4163.212 5529.829 7343.856 3459.507 4994.522 7217.714 10435.65 15089.50 21813.09 3151533 45497.17