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You have just purchased a car and taken out a $53,000 loan. The loan has a five-year term with monthly payments and an APR of 5.7%. a. How much will you pay in interest, and how much will you pay in principal, during the first month, second month, and first year? (Hint: Compute the loan balance after one month, two months, and one year.) b. How much will you pay in interest, and how much will you pay in principal, during the fourth year (i.e., between three and four years from now)? (Note: Be careful not to round any intermediate steps less than six decimal places.)
Expert Answer
| 1) | Interest | Principal | |||||||||
| First Month | $ 251.750000 | $ 765.511479 | |||||||||
| Second Month | $ 248.113820 | $ 769.147658 | |||||||||
| First Year | $ 2,777.171423 | $ 9,429.966322 | |||||||||
| 2) | Interest | Principal | |||||||||
| During fourth Year | $ 1,023.098643 | $ 11,184.039102 | |||||||||
| Worings: | |||||||||||
| Step-1:Calculate monthly payment | |||||||||||
| Monthly Payment | = | Loan Amount/Cumulative discount factor | |||||||||
| = | $ 53,000 | / | 52.100665 | ||||||||
| = | $ 1,017.261479 | ||||||||||
| Cumulative discount factor | = | (1-(1+i)^-n)/i | Where, | ||||||||
| = | (1-(1+0.00475)^-60)/0.00475 | i | = | 0.057/12 | = | 0.004750 | |||||
| = | 52.100665 | n | = | 5*12 | = | 60 | |||||
| Step-2:Prepare Loan Amortization Schedule | |||||||||||
| Loan Amortization schedule: | |||||||||||
| Months | Beginning Loan Amount | Monthly Interest | Monthly Payment | Principal Payment | Ending Loan Balance | ||||||
| a | b=a*5.7%/12 | c | d=c-b | a-d | |||||||
| 1 | $ 53,000.000000 | $ 251.750000 | $ 1,017.261479 | $ 765.511479 | $ 52,234.488521 | ||||||
| 2 | $ 52,234.488521 | $ 248.113820 | $ 1,017.261479 | $ 769.147658 | $ 51,465.340863 | ||||||
| 3 | $ 51,465.340863 | $ 244.460369 | $ 1,017.261479 | $ 772.801110 | $ 50,692.539753 | ||||||
| 4 | $ 50,692.539753 | $ 240.789564 | $ 1,017.261479 | $ 776.471915 | $ 49,916.067838 | ||||||
| 5 | $ 49,916.067838 | $ 237.101322 | $ 1,017.261479 | $ 780.160157 | $ 49,135.907682 | ||||||
| 6 | $ 49,135.907682 | $ 233.395561 | $ 1,017.261479 | $ 783.865917 | $ 48,352.041765 | ||||||
| 7 | $ 48,352.041765 | $ 229.672198 | $ 1,017.261479 | $ 787.589280 | $ 47,564.452484 | ||||||
| 8 | $ 47,564.452484 | $ 225.931149 | $ 1,017.261479 | $ 791.330329 | $ 46,773.122155 | ||||||
| 9 | $ 46,773.122155 | $ 222.172330 | $ 1,017.261479 | $ 795.089149 | $ 45,978.033006 | ||||||
| 10 | $ 45,978.033006 | $ 218.395657 | $ 1,017.261479 | $ 798.865822 | $ 45,179.167184 | ||||||
| 11 | $ 45,179.167184 | $ 214.601044 | $ 1,017.261479 | $ 802.660435 | $ 44,376.506750 | ||||||
| 12 | $ 44,376.506750 | $ 210.788407 | $ 1,017.261479 | $ 806.473072 | $ 43,570.033678 | ||||||
| Total | $ 2,777.171423 | $ 12,207.137745 | $ 9,429.966322 | ||||||||
| 13 | $ 43,570.033678 | $ 206.957660 | $ 1,017.261479 | $ 810.303819 | $ 42,759.729859 | ||||||
| 14 | $ 42,759.729859 | $ 203.108717 | $ 1,017.261479 | $ 814.152762 | $ 41,945.577097 | ||||||
| 15 | $ 41,945.577097 | $ 199.241491 | $ 1,017.261479 | $ 818.019988 | $ 41,127.557110 | ||||||
| 16 | $ 41,127.557110 | $ 195.355896 | $ 1,017.261479 | $ 821.905582 | $ 40,305.651527 | ||||||
| 17 | $ 40,305.651527 | $ 191.451845 | $ 1,017.261479 | $ 825.809634 | $ 39,479.841893 | ||||||
| 18 | $ 39,479.841893 | $ 187.529249 | $ 1,017.261479 | $ 829.732230 | $ 38,650.109664 | ||||||
| 19 | $ 38,650.109664 | $ 183.588021 | $ 1,017.261479 | $ 833.673458 | $ 37,816.436206 | ||||||
| 20 | $ 37,816.436206 | $ 179.628072 | $ 1,017.261479 | $ 837.633407 | $ 36,978.802799 | ||||||
| 21 | $ 36,978.802799 | $ 175.649313 | $ 1,017.261479 | $ 841.612165 | $ 36,137.190633 | ||||||
| 22 | $ 36,137.190633 | $ 171.651656 | $ 1,017.261479 | $ 845.609823 | $ 35,291.580810 | ||||||
| 23 | $ 35,291.580810 | $ 167.635009 | $ 1,017.261479 | $ 849.626470 | $ 34,441.954340 | ||||||
| 24 | $ 34,441.954340 | $ 163.599283 | $ 1,017.261479 | $ 853.662196 | $ 33,588.292145 | ||||||
| Total | $ 2,225.396212 | $ 12,207.137745 | $ 9,981.741533 | ||||||||
| 25 | $ 33,588.292145 | $ 159.544388 | $ 1,017.261479 | $ 857.717091 | $ 32,730.575054 | ||||||
| 26 | $ 32,730.575054 | $ 155.470232 | $ 1,017.261479 | $ 861.791247 | $ 31,868.783806 | ||||||
| 27 | $ 31,868.783806 | $ 151.376723 | $ 1,017.261479 | $ 865.884756 | $ 31,002.899051 | ||||||
| 28 | $ 31,002.899051 | $ 147.263770 | $ 1,017.261479 | $ 869.997708 | $ 30,132.901342 | ||||||
| 29 | $ 30,132.901342 | $ 143.131281 | $ 1,017.261479 | $ 874.130197 | $ 29,258.771145 | ||||||
| 30 | $ 29,258.771145 | $ 138.979163 | $ 1,017.261479 | $ 878.282316 | $ 28,380.488829 | ||||||
| 31 | $ 28,380.488829 | $ 134.807322 | $ 1,017.261479 | $ 882.454157 | $ 27,498.034672 | ||||||
| 32 | $ 27,498.034672 | $ 130.615665 | $ 1,017.261479 | $ 886.645814 | $ 26,611.388858 | ||||||
| 33 | $ 26,611.388858 | $ 126.404097 | $ 1,017.261479 | $ 890.857382 | $ 25,720.531477 | ||||||
| 34 | $ 25,720.531477 | $ 122.172525 | $ 1,017.261479 | $ 895.088954 | $ 24,825.442522 | ||||||
| 35 | $ 24,825.442522 | $ 117.920852 | $ 1,017.261479 | $ 899.340627 | $ 23,926.101896 | ||||||
| 36 | $ 23,926.101896 | $ 113.648984 | $ 1,017.261479 | $ 903.612495 | $ 23,022.489401 | ||||||
| Total | $ 1,641.335001 | $ 12,207.137745 | $ 10,565.802744 | ||||||||
| 37 | $ 23,022.489401 | $ 109.356825 | $ 1,017.261479 | $ 907.904654 | $ 22,114.584747 | ||||||
| 38 | $ 22,114.584747 | $ 105.044278 | $ 1,017.261479 | $ 912.217201 | $ 21,202.367546 | ||||||
| 39 | $ 21,202.367546 | $ 100.711246 | $ 1,017.261479 | $ 916.550233 | $ 20,285.817313 | ||||||
| 40 | $ 20,285.817313 | $ 96.357632 | $ 1,017.261479 | $ 920.903847 | $ 19,364.913466 | ||||||
| 41 | $ 19,364.913466 | $ 91.983339 | $ 1,017.261479 | $ 925.278140 | $ 18,439.635326 | ||||||
| 42 | $ 18,439.635326 | $ 87.588268 | $ 1,017.261479 | $ 929.673211 | $ 17,509.962115 | ||||||
| 43 | $ 17,509.962115 | $ 83.172320 | $ 1,017.261479 | $ 934.089159 | $ 16,575.872957 | ||||||
| 44 | $ 16,575.872957 | $ 78.735397 | $ 1,017.261479 | $ 938.526082 | $ 15,637.346875 | ||||||
| 45 | $ 15,637.346875 | $ 74.277398 | $ 1,017.261479 | $ 942.984081 | $ 14,694.362793 | ||||||
| 46 | $ 14,694.362793 | $ 69.798223 | $ 1,017.261479 | $ 947.463255 | $ 13,746.899538 | ||||||
| 47 | $ 13,746.899538 | $ 65.297773 | $ 1,017.261479 | $ 951.963706 | $ 12,794.935832 | ||||||
| 48 | $ 12,794.935832 | $ 60.775945 | $ 1,017.261479 | $ 956.485534 | $ 11,838.450298 | ||||||
| Total | $ 1,023.098643 | $ 12,207.137745 | $ 11,184.039102 | ||||||||
| 49 | $ 11,838.450298 | $ 56.232639 | $ 1,017.261479 | $ 961.028840 | $ 10,877.421459 | ||||||
| 50 | $ 10,877.421459 | $ 51.667752 | $ 1,017.261479 | $ 965.593727 | $ 9,911.827732 | ||||||
| 51 | $ 9,911.827732 | $ 47.081182 | $ 1,017.261479 | $ 970.180297 | $ 8,941.647435 | ||||||
| 52 | $ 8,941.647435 | $ 42.472825 | $ 1,017.261479 | $ 974.788653 | $ 7,966.858781 | ||||||
| 53 | $ 7,966.858781 | $ 37.842579 | $ 1,017.261479 | $ 979.418900 | $ 6,987.439882 | ||||||
| 54 | $ 6,987.439882 | $ 33.190339 | $ 1,017.261479 | $ 984.071139 | $ 6,003.368743 | ||||||
| 55 | $ 6,003.368743 | $ 28.516002 | $ 1,017.261479 | $ 988.745477 | $ 5,014.623265 | ||||||
| 56 | $ 5,014.623265 | $ 23.819461 | $ 1,017.261479 | $ 993.442018 | $ 4,021.181247 | ||||||
| 57 | $ 4,021.181247 | $ 19.100611 | $ 1,017.261479 | $ 998.160868 | $ 3,023.020379 | ||||||
| 58 | $ 3,023.020379 | $ 14.359347 | $ 1,017.261479 | $ 1,002.902132 | $ 2,020.118247 | ||||||
| 59 | $ 2,020.118247 | $ 9.595562 | $ 1,017.261479 | $ 1,007.665917 | $ 1,012.452330 | ||||||
| 60 | $ 1,012.452330 | $ 4.809149 | $ 1,017.261479 | $ 1,012.452330 | $ 0.000000 | ||||||
| Total | $ 368.687447 | $ 12,207.137745 | $ 11,838.450298 | ||||||||
| Sub-Total | $ 8,035.688725 | $ 61,035.688725 | $ 53,000.000000 | ||||||||
