Sarah started working as an investment banker with a starting annual salary of $

Question

Sarah started working as an investment banker with a starting annual salary of $84,000 on January 1, 2014. She receives her salary in equal monthly instalments at the end of each month. She expects to receive a 4% raise every year until she retires after 30 years of hard work (i.e. she will receive monthly paycheques of $7,000 at the end of each month in 2014, $7,000 × 1.04 at the end of every month in 2015, and so on). As part of her benefits package her employer offers to put in 5% of her monthly paycheque into a retirement savings account as long as she puts in an equal amount at the same time (so if she puts $350 = 0.05 × $7,000 into the retirement account form her own salary at the end of this month, her employer will match that amount by an additional $350 so in total, $700 will be deposited into her retirement account on January 31, 2014). Assuming she will continue to contribute exactly 5% every month for 30 years, how much money will Sarah have in her retirement account by the time she retires? (Her last work day and paycheque will fall on December 31st, so assume she and her employer make one last payment into the account and then she checks her balance.) Sarah earns 6.6% (APR, compounded monthly) interest on funds invested in her retirement account.

show all work please

step by step

do not use excel please

explain the question with words not just numbers

thanks in advance

Explanation / Answer

Sarah invests 5% of her salary in the retirement account and her employer also adds up the same amount in the account. Hence the amount that gets deposited every month is actually 10% of the salary that month. Sorry but have to excel for such a huge calculation. I will explain each and every number that I got in the table and how I got it.

There are few things to elaborate in this table

First is the interest rate that has been taken on a monthly basis. As the annuity is getting compounded monthly we have to calculate the monthly rate that will be applicable to the balance in the retirement account. Hence the monthly interest rate will be 6.6% divided by 12 that give us 0.55%.

Second is the balance on every end of month. As the deposit is made at the end of month the interest amount gets accumulated at the end of next month. Hence the balance on the first month was the same as the deposit as it was done on the last day of the month. The interest amount has been calculated on the balance in the retirement account. And the balance is calculated by adding the interest amount accumulated for the previous month and the deposit made that month. For example for the second month the balance is 1403.85 that has been calculated by adding the interest amount accumulated on the balance of previous month 700 that comes to 3.85 and the deposit made till now that is 1400 (700 for previous month and 700 for this month).

End of Month Monthly Salary Yearly Package Deposit Monthly Interest Rate Month End Balance 1 7000 84000 700 0.55% 700 2 7000 84000 700 1403.85 3 7000 84000 700 2111.57 4 7000 84000 700 2823.18 5 7000 84000 700 3538.71 6 7000 84000 700 4258.18 7 7000 84000 700 4981.60 8 7000 84000 700 5708.99 9 7000 84000 700 6440.39 10 7000 84000 700 7175.82 11 7000 84000 700 7915.28 12 7000 84000 700 8658.82 13 7280 87360 728 9434.44 14 7280 87360 728 10214.33 15 7280 87360 728 10998.51 16 7280 87360 728 11787.00 17 7280 87360 728 12579.83 18 7280 87360 728 13377.02 19 7280 87360 728 14178.59 20 7280 87360 728 14984.57 21 7280 87360 728 15794.99 22 7280 87360 728 16609.86 23 7280 87360 728 17429.22 24 7280 87360 728 18253.08 25 7571.2 90854.4 757.12 19110.59 26 7571.2 90854.4 757.12 19972.82 27 7571.2 90854.4 757.12 20839.79 28 7571.2 90854.4 757.12 21711.53 29 7571.2 90854.4 757.12 22588.06 30 7571.2 90854.4 757.12 23469.41 31 7571.2 90854.4 757.12 24355.62 32 7571.2 90854.4 757.12 25246.69 33 7571.2 90854.4 757.12 26142.67 34 7571.2 90854.4 757.12 27043.57 35 7571.2 90854.4 757.12 27949.43 36 7571.2 90854.4 757.12 28860.27 37 7874.05 94488.58 787.40 29806.41 38 7874.05 94488.58 787.40 30757.75 39 7874.05 94488.58 787.40 31714.32 40 7874.05 94488.58 787.40 32676.16 41 7874.05 94488.58 787.40 33643.28 42 7874.05 94488.58 787.40 34615.72 43 7874.05 94488.58 787.40 35593.51 44 7874.05 94488.58 787.40 36576.68 45 7874.05 94488.58 787.40 37565.26 46 7874.05 94488.58 787.40 38559.27 47 7874.05 94488.58 787.40 39558.75 48 7874.05 94488.58 787.40 40563.73 49 8189.01 98268.12 818.90 41605.73 50 8189.01 98268.12 818.90 42653.47 51 8189.01 98268.12 818.90 43706.96 52 8189.01 98268.12 818.90 44766.25 53 8189.01 98268.12 818.90 45831.37 54 8189.01 98268.12 818.90 46902.34 55 8189.01 98268.12 818.90 47979.20 56 8189.01 98268.12 818.90 49061.99 57 8189.01 98268.12 818.90 50150.73 58 8189.01 98268.12 818.90 51245.46 59 8189.01 98268.12 818.90 52346.21 60 8189.01 98268.12 818.90 53453.02 61 8516.57 102198.84 851.66 54598.67 62 8516.57 102198.84 851.66 55750.62 63 8516.57 102198.84 851.66 56908.90 64 8516.57 102198.84 851.66 58073.56 65 8516.57 102198.84 851.66 59244.62 66 8516.57 102198.84 851.66 60422.12 67 8516.57 102198.84 851.66 61606.10 68 8516.57 102198.84 851.66 62796.59 69 8516.57 102198.84 851.66 63993.63 70 8516.57 102198.84 851.66 65197.25 71 8516.57 102198.84 851.66 66407.49 72 8516.57 102198.84 851.66 67624.39 73 8857.23 106286.80 885.72 68882.05 74 8857.23 106286.80 885.72 70146.62 75 8857.23 106286.80 885.72 71418.15 76 8857.23 106286.80 885.72 72696.68 77 8857.23 106286.80 885.72 73982.23 78 8857.23 106286.80 885.72 75274.86 79 8857.23 106286.80 885.72 76574.59 80 8857.23 106286.80 885.72 77881.48 81 8857.23 106286.80 885.72 79195.55 82 8857.23 106286.80 885.72 80516.85 83 8857.23 106286.80 885.72 81845.41 84 8857.23 106286.80 885.72 83181.29 85 9211.52 110538.27 921.15 84559.93 86 9211.52 110538.27 921.15 85946.17 87 9211.52 110538.27 921.15 87340.02 88 9211.52 110538.27 921.15 88741.54 89 9211.52 110538.27 921.15 90150.78 90 9211.52 110538.27 921.15 91567.76 91 9211.52 110538.27 921.15 92992.53 92 9211.52 110538.27 921.15 94425.14 93 9211.52 110538.27 921.15 95865.63 94 9211.52 110538.27 921.15 97314.05 95 9211.52 110538.27 921.15 98770.43 96 9211.52 110538.27 921.15 100234.82 97 9579.98 114959.80 958.00 101744.11 98 9579.98 114959.80 958.00 103261.70 99 9579.98 114959.80 958.00 104787.63 100 9579.98 114959.80 958.00 106321.96 101 9579.98 114959.80 958.00 107864.73 102 9579.98 114959.80 958.00 109415.99 103 9579.98 114959.80 958.00 110975.77 104 9579.98 114959.80 958.00 112544.14 105 9579.98 114959.80 958.00 114121.13 106 9579.98 114959.80 958.00 115706.80 107 9579.98 114959.80 958.00 117301.18 108 9579.98 114959.80 958.00 118904.34 109 9963.18 119558.19 996.32 120554.63 110 9963.18 119558.19 996.32 122214.00 111 9963.18 119558.19 996.32 123882.49 112 9963.18 119558.19 996.32 125560.16 113 9963.18 119558.19 996.32 127247.06 114 9963.18 119558.19 996.32 128943.24 115 9963.18 119558.19 996.32 130648.75 116 9963.18 119558.19 996.32 132363.63 117 9963.18 119558.19 996.32 134087.95 118 9963.18 119558.19 996.32 135821.75 119 9963.18 119558.19 996.32 137565.09 120 9963.18 119558.19 996.32 139318.02 121 10361.71 124340.52 1036.17 141120.44 122 10361.71 124340.52 1036.17 142932.77 123 10361.71 124340.52 1036.17 144755.07 124 10361.71 124340.52 1036.17 146587.40 125 10361.71 124340.52 1036.17 148429.80 126 10361.71 124340.52 1036.17 150282.33 127 10361.71 124340.52 1036.17 152145.06 128 10361.71 124340.52 1036.17 154018.02 129 10361.71 124340.52 1036.17 155901.30 130 10361.71 124340.52 1036.17 157794.92 131 10361.71 124340.52 1036.17 159698.97 132 10361.71 124340.52 1036.17 161613.48 133 10776.18 129314.14 1077.62 163579.97 134 10776.18 129314.14 1077.62 165557.28 135 10776.18 129314.14 1077.62 167545.46 136 10776.18 129314.14 1077.62 169544.58 137 10776.18 129314.14 1077.62 171554.70 138 10776.18 129314.14 1077.62 173575.86 139 10776.18 129314.14 1077.62 175608.15 140 10776.18 129314.14 1077.62 177651.61 141 10776.18 129314.14 1077.62 179706.31 142 10776.18 129314.14 1077.62 181772.32 143 10776.18 129314.14 1077.62 183849.68 144 10776.18 129314.14 1077.62 185938.47 145 11207.23 134486.71 1120.72 188081.86 146 11207.23 134486.71 1120.72 190237.03 147 11207.23 134486.71 1120.72 192404.06 148 11207.23 134486.71 1120.72 194583.00 149 11207.23 134486.71 1120.72 196773.93 150 11207.23 134486.71 1120.72 198976.91 151 11207.23 134486.71 1120.72 201192.00 152 11207.23 134486.71 1120.72 203419.28 153 11207.23 134486.71 1120.72 205658.81 154 11207.23 134486.71 1120.72 207910.66 155 11207.23 134486.71 1120.72 210174.89 156 11207.23 134486.71 1120.72 212451.57 157 11655.51 139866.17 1165.55 214785.61 158 11655.51 139866.17 1165.55 217132.48 159 11655.51 139866.17 1165.55 219492.26 160 11655.51 139866.17 1165.55 221865.02 161 11655.51 139866.17 1165.55 224250.83 162 11655.51 139866.17 1165.55 226649.76 163 11655.51 139866.17 1165.55 229061.88 164 11655.51 139866.17 1165.55 231487.28 165 11655.51 139866.17 1165.55 233926.01 166 11655.51 139866.17 1165.55 236378.15 167 11655.51 139866.17 1165.55 238843.78 168 11655.51 139866.17 1165.55 241322.98 169 12121.74 145460.82 1212.17 243862.43 170 12121.74 145460.82 1212.17 246415.84 171 12121.74 145460.82 1212.17 248983.30 172 12121.74 145460.82 1212.17 251564.88 173 12121.74 145460.82 1212.17 254160.67 174 12121.74 145460.82 1212.17 256770.72 175 12121.74 145460.82 1212.17 259395.13 176 12121.74 145460.82 1212.17 262033.98 177 12121.74 145460.82 1212.17 264687.34 178 12121.74 145460.82 1212.17 267355.30 179 12121.74 145460.82 1212.17 270037.92 180 12121.74 145460.82 1212.17 272735.31 181 12606.60 151279.25 1260.66 275496.01 182 12606.60 151279.25 1260.66 278271.90 183 12606.60 151279.25 1260.66 281063.05 184 12606.60 151279.25 1260.66 283869.56 185 12606.60 151279.25 1260.66 286691.51 186 12606.60 151279.25 1260.66 289528.97 187 12606.60 151279.25 1260.66 292382.04 188 12606.60 151279.25 1260.66 295250.80 189 12606.60 151279.25 1260.66 298135.34 190 12606.60 151279.25 1260.66 301035.74 191 12606.60 151279.25 1260.66 303952.10 192 12606.60 151279.25 1260.66 306884.50 193 13110.87 157330.42 1311.09 309883.45 194 13110.87 157330.42 1311.09 312898.90 195 13110.87 157330.42 1311.09 315930.93 196 13110.87 157330.42 1311.09 318979.63 197 13110.87 157330.42 1311.09 322045.11 198 13110.87 157330.42 1311.09 325127.44 199 13110.87 157330.42 1311.09 328226.73 200 13110.87 157330.42 1311.09 331343.07 201 13110.87 157330.42 1311.09 334476.54 202 13110.87 157330.42 1311.09 337627.25 203 13110.87 157330.42 1311.09 340795.28 204 13110.87 157330.42 1311.09 343980.75 205 13635.30 163623.64 1363.53 347236.17 206 13635.30 163623.64 1363.53 350509.50 207 13635.30 163623.64 1363.53 353800.83 208 13635.30 163623.64 1363.53 357110.27 209 13635.30 163623.64 1363.53 360437.90 210 13635.30 163623.64 1363.53 363783.84 211 13635.30 163623.64 1363.53 367148.18 212 13635.30 163623.64 1363.53 370531.03 213 13635.30 163623.64 1363.53 373932.48 214 13635.30 163623.64 1363.53 377352.64 215 13635.30 163623.64 1363.53 380791.61 216 13635.30 163623.64 1363.53 384249.49 217 14180.72 170168.59 1418.07 387780.94 218 14180.72 170168.59 1418.07 391331.80 219 14180.72 170168.59 1418.07 394902.20 220 14180.72 170168.59 1418.07 398492.23 221 14180.72 170168.59 1418.07 402102.01 222 14180.72 170168.59 1418.07 405731.64 223 14180.72 170168.59 1418.07 409381.24 224 14180.72 170168.59 1418.07 413050.91 225 14180.72 170168.59 1418.07 416740.76 226 14180.72 170168.59 1418.07 420450.91 227 14180.72 170168.59 1418.07 424181.46 228 14180.72 170168.59 1418.07 427932.53 229 14747.94 176975.33 1474.79 431760.95 230 14747.94 176975.33 1474.79 435610.43 231 14747.94 176975.33 1474.79 439481.08 232 14747.94 176975.33 1474.79 443373.02 233 14747.94 176975.33 1474.79 447286.37 234 14747.94 176975.33 1474.79 451221.24 235 14747.94 176975.33 1474.79 455177.75 236 14747.94 176975.33 1474.79 459156.02 237 14747.94 176975.33 1474.79 463156.17 238 14747.94 176975.33 1474.79 467178.33 239 14747.94 176975.33 1474.79 471222.60 240 14747.94 176975.33 1474.79 475289.12 241 15337.86 184054.34 1533.79 479437.00 242 15337.86 184054.34 1533.79 483607.69 243 15337.86 184054.34 1533.79 487801.32 244 15337.86 184054.34 1533.79 492018.01 245 15337.86 184054.34 1533.79 496257.89 246 15337.86 184054.34 1533.79 500521.10 247 15337.86 184054.34 1533.79 504807.75 248 15337.86 184054.34 1533.79 509117.98 249 15337.86 184054.34 1533.79 513451.92 250 15337.86 184054.34 1533.79 517809.69 251 15337.86 184054.34 1533.79 522191.43 252 15337.86 184054.34 1533.79 526597.27 253 15951.38 191416.52 1595.14 531088.69 254 15951.38 191416.52 1595.14 535604.81 255 15951.38 191416.52 1595.14 540145.78 256 15951.38 191416.52 1595.14 544711.72 257 15951.38 191416.52 1595.14 549302.77 258 15951.38 191416.52 1595.14 553919.07 259 15951.38 191416.52 1595.14 558560.76 260 15951.38 191416.52 1595.14 563227.99 261 15951.38 191416.52 1595.14 567920.88 262 15951.38 191416.52 1595.14 572639.58 263 15951.38 191416.52 1595.14 577384.24 264 15951.38 191416.52 1595.14 582154.99 265 16589.43 199073.18 1658.94 587015.78 266 16589.43 199073.18 1658.94 591903.31 267 16589.43 199073.18 1658.94 596817.72 268 16589.43 199073.18 1658.94 601759.16 269 16589.43 199073.18 1658.94 606727.78 270 16589.43 199073.18 1658.94 611723.73 271 16589.43 199073.18 1658.94 616747.15 272 16589.43 199073.18 1658.94 621798.20 273 16589.43 199073.18 1658.94 626877.04 274 16589.43 199073.18 1658.94 631983.81 275 16589.43 199073.18 1658.94 637118.66 276 16589.43 199073.18 1658.94 642281.75 277 17253.01 207036.11 1725.30 647539.61 278 17253.01 207036.11 1725.30 652826.37 279 17253.01 207036.11 1725.30 658142.22 280 17253.01 207036.11 1725.30 663487.30 281 17253.01 207036.11 1725.30 668861.78 282 17253.01 207036.11 1725.30 674265.82 283 17253.01 207036.11 1725.30 679699.59 284 17253.01 207036.11 1725.30 685163.24 285 17253.01 207036.11 1725.30 690656.94 286 17253.01 207036.11 1725.30 696180.85 287 17253.01 207036.11 1725.30 701735.14 288 17253.01 207036.11 1725.30 707319.99 289 17943.13 215317.55 1794.31 713004.56 290 17943.13 215317.55 1794.31 718720.40 291 17943.13 215317.55 1794.31 724467.67 292 17943.13 215317.55 1794.31 730246.56 293 17943.13 215317.55 1794.31 736057.23 294 17943.13 215317.55 1794.31 741899.86 295 17943.13 215317.55 1794.31 747774.62 296 17943.13 215317.55 1794.31 753681.69 297 17943.13 215317.55 1794.31 759621.25 298 17943.13 215317.55 1794.31 765593.48 299 17943.13 215317.55 1794.31 771598.56 300 17943.13 215317.55 1794.31 777636.67 301 18660.85 223930.25 1866.09 783779.75 302 18660.85 223930.25 1866.09 789956.63 303 18660.85 223930.25 1866.09 796167.47 304 18660.85 223930.25 1866.09 802412.48 305 18660.85 223930.25 1866.09 808691.83 306 18660.85 223930.25 1866.09 815005.73 307 18660.85 223930.25 1866.09 821354.34 308 18660.85 223930.25 1866.09 827737.88 309 18660.85 223930.25 1866.09 834156.52 310 18660.85 223930.25 1866.09 840610.47 311 18660.85 223930.25 1866.09 847099.91 312 18660.85 223930.25 1866.09 853625.04 313 19407.29 232887.46 1940.73 860260.71 314 19407.29 232887.46 1940.73 866932.87 315 19407.29 232887.46 1940.73 873641.73 316 19407.29 232887.46 1940.73 880387.49 317 19407.29 232887.46 1940.73 887170.35 318 19407.29 232887.46 1940.73 893990.52 319 19407.29 232887.46 1940.73 900848.19 320 19407.29 232887.46 1940.73 907743.59 321 19407.29 232887.46 1940.73 914676.91 322 19407.29 232887.46 1940.73 921648.36 323 19407.29 232887.46 1940.73 928658.15 324 19407.29 232887.46 1940.73 935706.50 325 20183.58 242202.96 2018.36 942871.25 326 20183.58 242202.96 2018.36 950075.40 327 20183.58 242202.96 2018.36 957319.17 328 20183.58 242202.96 2018.36 964602.78 329 20183.58 242202.96 2018.36 971926.46 330 20183.58 242202.96 2018.36 979290.41 331 20183.58 242202.96 2018.36 986694.86 332 20183.58 242202.96 2018.36 994140.04 333 20183.58 242202.96 2018.36 1001626.17 334 20183.58 242202.96 2018.36 1009153.47 335 20183.58 242202.96 2018.36 1016722.18 336 20183.58 242202.96 2018.36 1024332.51 337 20990.92 251891.08 2099.09 1032065.43 338 20990.92 251891.08 2099.09 1039840.88 339 20990.92 251891.08 2099.09 1047659.10 340 20990.92 251891.08 2099.09 1055520.31 341 20990.92 251891.08 2099.09 1063424.77 342 20990.92 251891.08 2099.09 1071372.70 343 20990.92 251891.08 2099.09 1079364.34 344 20990.92 251891.08 2099.09 1087399.93 345 20990.92 251891.08 2099.09 1095479.73 346 20990.92 251891.08 2099.09 1103603.96 347 20990.92 251891.08 2099.09 1111772.87 348 20990.92 251891.08 2099.09 1119986.71 349 21830.56 261966.72 2183.06 1128329.70 350 21830.56 261966.72 2183.06 1136718.57 351 21830.56 261966.72 2183.06 1145153.58 352 21830.56 261966.72 2183.06 1153634.98 353 21830.56 261966.72 2183.06 1162163.02 354 21830.56 261966.72 2183.06 1170737.98 355 21830.56 261966.72 2183.06 1179360.09 356 21830.56 261966.72 2183.06 1188029.63 357 21830.56 261966.72 2183.06 1196746.85 358 21830.56 261966.72 2183.06 1205512.01 359 21830.56 261966.72 2183.06 1214325.38 360 21830.56 261966.72 2183.06 1223187.23

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