Question

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A loan is being repaid by the amortization method with an installment at the end of...

A loan is being repaid by the amortization method with an installment at the end of each 80 quarters at 6% annual effective interest, the first payment one quarter after the loan is made. In which payment are the principal and the interest most nearly equal to each other?

ANS: 33 Please show the steps.

Solutions

Expert Solution

Effective Interest Rate = [(1+Interest Rate)^Number of times Compounded in a year]-1

0.06 = [(1+i)^4]-1

1.06^1/4 = 1+i

Therefore, Quarterly Interest = i = 1.01467-1 = 0.01467 = 1.467%

Lets Assume a loan of $100 is borrowed,

EMI = P*i*(1+i)^n/[{(1+i)^n}-1]

Where,

P = Principal = 100

i= Interest Rate = 0.01467

n= Number of periods = 80

Therefore, EMI = 100*0.01467*(1+0.01467)^80/[{(1+0.01467)^80}-1]

= 1.467*(3.20616)/[3.20616-1] = 4.70344/2.20616 = $2.131958

Amortization Schedule will be:

Period Opening Principal
(previous closing)		 Interest
(opening*0.01467)		 Installment Principal Repayment
(installment-interest)		 Closing Principal
(opening-principal repayment)
1 100 1.467 2.131958 0.664958 99.335042
2 99.335042 1.457245066 2.131958 0.67471293 98.6603291
3 98.6603291 1.447347027 2.131958 0.68461097 97.9757181
4 97.9757181 1.437303784 2.131958 0.69465422 97.2810639
5 97.2810639 1.427113207 2.131958 0.70484479 96.5762191
6 96.5762191 1.416773134 2.131958 0.71518487 95.8610342
7 95.8610342 1.406281372 2.131958 0.72567663 95.1353576
8 95.1353576 1.395635696 2.131958 0.7363223 94.3990353
9 94.3990353 1.384833848 2.131958 0.74712415 93.6519111
10 93.6519111 1.373873536 2.131958 0.75808446 92.8938267
11 92.8938267 1.362752437 2.131958 0.76920556 92.1246211
12 92.1246211 1.351468192 2.131958 0.78048981 91.3441313
13 91.3441313 1.340018406 2.131958 0.79193959 90.5521917
14 90.5521917 1.328400652 2.131958 0.80355735 89.7486344
15 89.7486344 1.316612466 2.131958 0.81534553 88.9332888
16 88.9332888 1.304651347 2.131958 0.82730665 88.1059822
17 88.1059822 1.292514758 2.131958 0.83944324 87.2665389
18 87.2665389 1.280200126 2.131958 0.85175787 86.4147811
19 86.4147811 1.267704838 2.131958 0.86425316 85.5505279
20 85.5505279 1.255026244 2.131958 0.87693176 84.6735961
21 84.6735961 1.242161655 2.131958 0.88979634 83.7837998
22 83.7837998 1.229108343 2.131958 0.90284966 82.8809501
23 82.8809501 1.215863539 2.131958 0.91609446 81.9648557
24 81.9648557 1.202424433 2.131958 0.92953357 81.0353221
25 81.0353221 1.188788175 2.131958 0.94316982 80.0921523
26 80.0921523 1.174951874 2.131958 0.95700613 79.1351462
27 79.1351462 1.160912594 2.131958 0.97104541 78.1641008
28 78.1641008 1.146667358 2.131958 0.98529064 77.1788101
29 77.1788101 1.132213144 2.131958 0.99974486 76.1790653
30 76.1790653 1.117546887 2.131958 1.01441111 75.1646541
31 75.1646541 1.102665476 2.131958 1.02929252 74.1353616
32 74.1353616 1.087565755 2.131958 1.04439225 73.0909694
33 73.0909694 1.072244521 2.131958 1.05971348 72.0312559
34 72.0312559 1.056698524 2.131958 1.07525948 70.9559964
35 70.9559964 1.040924467 2.131958 1.09103353 69.8649629
36 69.8649629 1.024919006 2.131958 1.10703899 68.7579239
37 68.7579239 1.008678743 2.131958 1.12327926 67.6346446
38 67.6346446 0.992200237 2.131958 1.13975776 66.4948869
39 66.4948869 0.97547999 2.131958 1.15647801 65.3384089
40 65.3384089 0.958514458 2.131958 1.17344354 64.1649653
41 64.1649653 0.941300041 2.131958 1.19065796 62.9743074
42 62.9743074 0.923833089 2.131958 1.20812491 61.7661824
43 61.7661824 0.906109897 2.131958 1.2258481 60.5403343
44 60.5403343 0.888126705 2.131958 1.2438313 59.296503
45 59.296503 0.8698797 2.131958 1.2620783 58.0344247
46 58.0344247 0.851365011 2.131958 1.28059299 56.7538318
47 56.7538318 0.832578712 2.131958 1.29937929 55.4544525
48 55.4544525 0.813516818 2.131958 1.31844118 54.1360113
49 54.1360113 0.794175286 2.131958 1.33778271 52.7982286
50 52.7982286 0.774550013 2.131958 1.35740799 51.4408206
51 51.4408206 0.754636838 2.131958 1.37732116 50.0634994
52 50.0634994 0.734431537 2.131958 1.39752646 48.665973
53 48.665973 0.713929823 2.131958 1.41802818 47.2479448
54 47.2479448 0.69312735 2.131958 1.43883065 45.8091141
55 45.8091141 0.672019704 2.131958 1.4599383 44.3491758
56 44.3491758 0.65060241 2.131958 1.48135559 42.8678203
57 42.8678203 0.628870923 2.131958 1.50308708 41.3647332
58 41.3647332 0.606820636 2.131958 1.52513736 39.8395958
59 39.8395958 0.584446871 2.131958 1.54751113 38.2920847
60 38.2920847 0.561744882 2.131958 1.57021312 36.7218716
61 36.7218716 0.538709856 2.131958 1.59324814 35.1286234
62 35.1286234 0.515336906 2.131958 1.61662109 33.5120023
63 33.5120023 0.491621074 2.131958 1.64033693 31.8716654
64 31.8716654 0.467557331 2.131958 1.66440067 30.2072647
65 30.2072647 0.443140574 2.131958 1.68881743 28.5184473
66 28.5184473 0.418365622 2.131958 1.71359238 26.8048549
67 26.8048549 0.393227222 2.131958 1.73873078 25.0661241
68 25.0661241 0.367720041 2.131958 1.76423796 23.3018862
69 23.3018862 0.34183867 2.131958 1.79011933 21.5117669
70 21.5117669 0.31557762 2.131958 1.81638038 19.6953865
71 19.6953865 0.28893132 2.131958 1.84302668 17.8523598
72 17.8523598 0.261894118 2.131958 1.87006388 15.9822959
73 15.9822959 0.234460281 2.131958 1.89749772 14.0847982
74 14.0847982 0.20662399 2.131958 1.92533401 12.1594642
75 12.1594642 0.17837934 2.131958 1.95357866 10.2058855
76 10.2058855 0.149720341 2.131958 1.98223766 8.22364787
77 8.22364787 0.120640914 2.131958 2.01131709 6.21233078
78 6.21233078 0.091134893 2.131958 2.04082311 4.17150767
79 4.17150767 0.061196018 2.131958 2.07076198 2.10074569
80 2.10074569 0.030817939 2.13156363 2.10074569 0

It is clear from the Amortization Schedule that Interest and Principal Repayment is almost same in 33rd payment

Therefore, 33rd Payment

jeff jeffy answered 2 years ago
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