Question

Use the following information regarding your retirement planning:

• You plan to work (and save) for 35 years, then retire (and spend money from your retirement account) for 25 years. After these 60 years, you expect that your retirement saving account will have $50,000 left to give to your family.
• You plan to save $4,000 in year 1, and you will increase this amount by 3% a year
• You want your retirement spending to increase by 2% per year
• You expect to earn a rate of return of 8% during your working years and 4.50% during retirement

To solve this problem, find the amount that you spend your first year of retirement.

Answer 1

It has been assumed that first investment   and first spending after retirement is done at year end. Future value of growth annuity FV = P[{(1+r)^n-(1-g)^n}/r-g] Where P=first payment=$4000 r=rate per period=8% g=growth rate=3% n=no.of period=35 years FV = $4000[{(1+0.08)^35-(1-0.03)^35}/0.08-0.03] = $4000[{14.7853-2.8139}/0.05] = $4000[11.9715/.05] = $4000*239.4296 = $ 957,718.40 Amount in retirement fund after 35 years =$957,718.4 Now his account will have $50,000 after 25 years from retirement to give to his family PV of those $50000 at the time of retirement                                        = = $50,000*PVAF(4.5%,25 years) = $50,000*0.3327 $16,635.00 PV of money which has been spend after retirement in 25 years at the time of retirement =$957,718.4-$16635=$941083.4 Present value of growing annuity PV = [P/r-g]*[1-{(1+g)/(1+r)}^n] where P=first payment r=rate per period=4.5% g=growth rate=2% n=no.of periods=25 years 941,083.40 = [P/0.045-0.02]*[1-{(1+0.02)/(1+0.045)}^25 941,083.40                = [P/0.025]*[1-{1.02/1.045}^25 941,083.40                = [P/0.025]*[1-{0.976077}^25] 941,083.40                = [P/0.025]*[1-0.545886] 941,083.40                = [p/0.025]*0.454114 941083.4/0.454114       = P/0.025 2,072,350.56 = P/0.025 2072350.56*.025 P $51,808.76 = P First payment =$51808.76 It has been assumed that first investment of $4,000 and first spending of $51808 is done at year end. There might be slight variation due to decimal places.Please do not downvote on that basis. Please upvote Verification amount Invested for 35 years amount spend for 25 years (a) (b) (.c) (d)=a*0.08 (.e)=B+C+D (a) (b) (.c) (d)=a*0.045 (.e)=B-C+D year op.bal amt interest @8% closing bal year op. bal amt interest closing bal 1 4000 4000 1 957718 51808 43097.31 949007.31 2 4000 4120 320 8440 2 949007.3 52844.16 42705.32895 938868.479 3 8440 4243.6 675.2 13358.8 3 938868.5 53901.04 42249.08155 927216.517 4 13358.8 4370.908 1068.704 18798.412 4 927216.5 54979.06 41724.74328 913962.197 5 18798.412 4502.035 1503.87296 24804.32 5 913962.2 56078.65 41128.29884 899011.85 6 24804.3202 4637.096 1984.345616 31425.762 6 899011.9 57200.22 40455.53325 882267.165 7 31425.76211 4776.209 2514.060969 38716.032 7 882267.2 58344.22 39702.02243 863624.965 8 38716.03227 4919.495 3097.282581 46732.81 8 863625 59511.11 38863.12342 842976.981 9 46732.81031 5067.08 3738.624825 55538.515 9 842977 60701.33 37933.96415 820209.616 10 55538.51546 5219.093 4443.081237 65200.689 10 820209.6 61915.36 36909.43273 795203.693 11 65200.68943 5375.666 5216.055155 75792.41 11 795203.7 63153.66 35784.16619 767834.196 12 75792.41011 5536.935 6063.392809 87392.738 12 767834.2 64416.74 34552.53883 737969.999 13 87392.7384 5703.044 6991.419072 100087.2 13 737970 65705.07 33208.64995 705473.578 14 100087.201 5874.135 8006.976081 113968.31 14 705473.6 67019.17 31746.31101 670200.717 15 113968.312 6050.359 9117.464956 129136.14 15 670200.7 68359.56 30159.03225 632000.193 16 129136.1358 6231.87 10330.89086 145698.9 16 632000.2 69726.75 28440.0087 590713.455 17 145698.8963 6418.826 11655.91171 163773.63 17 590713.5 71121.28 26582.10548 546174.279 18 163773.6338 6611.391 13101.8907 183486.92 18 546174.3 72543.71 24577.84254 498208.414 19 183486.915 6809.732 14678.9532 204975.6 19 498208.4 73994.58 22419.37862 446633.211 20 204975.6005 7014.024 16398.04804 228387.67 20 446633.2 75474.47 20098.49449 391257.232 21 228387.6727 7224.445 18271.01382 253883.13 21 391257.2 76983.96 17606.57545 331879.845 22 253883.1315 7441.178 20310.65052 281634.96 22 331879.8 78523.64 14934.59302 268290.796 23 281634.9603 7664.414 22530.79682 311830.17 23 268290.8 80094.11 12073.08582 200269.767 24 311830.1708 7894.346 24946.41366 344670.93 24 200269.8 81696 9012.139514 127585.909 25 344670.9305 8131.176 27573.67444 380375.78 25 127585.9 83329.92 5741.365924 $50,000 26 380375.7813 8375.112 30430.06251 419180.96 27 419180.9555 8626.365 33534.47644 461341.8 28 461341.7971 8885.156 36907.34377 507134.3 29 507134.2969 9151.711 40570.74375 556856.75 30 556856.7513 9426.262 44548.5401 610831.55 31 610831.5534 9709.05 48866.52427 669407.13 32 669407.1276 10000.32 53552.57021 732960.02 33 732960.0192 10300.33 58636.80153 801897.15 34 801897.1517 10609.34 64151.77214 876658.26 35 876658.2648 10927.62 70132.66119 957718.55