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Home > Algebra calculators > Amortization example
Amortization example ( Enter your problem )
( Enter your problem )
  1. Example : Find Monthly payment (A)
  2. Example : Loan amount (P)
  3. Example : Find Interest rate (i)
  4. Example : Find Loan term (n)
2. Example : Loan amount (P)
(Previous example)		4. Example : Find Loan term (n)
(Next example)

3. Example : Find Interest rate (i)


1. Find Interest Rate i = ?
for Monthly payment A = 126.63, Loan amount P = 12000, Loan term n = 10 years,

Solution:
`P=12000` (Loan amount)

`n=10` years (Number of periods)

Monthly Amount `A=126.63` per month

`n=10*12=120` months

`A=(P*i)/(1-(1+i)^-n)`

`:.126.63=(12000*i)/(1-(1+i)^-120)`

`:.(1-(1+i)^-120)/(i)=(12000)/(126.63)`

`:.(1-(1+i)^-120)/(i)=94.7643`

Now, find one solution using Newton Raphson method

Here `1-(1+x)^-120=94.7643x`

`:.-94.7643x+1-1/((1+x)^120)=0`

Let `f(x) = -94.7643x+1-1/((1+x)^120)`

`d/(dx)(-94.7643x+1-1/((1+x)^120))=-94.7643+120/((1+x)^121)`


`d/(dx)(-94.7643x+1-1/((1+x)^120))`

`=-d/(dx)(94.7643x)+d/(dx)(1)-d/(dx)(1/((1+x)^120))`

`d/(dx)(1/((1+x)^120))=-120/((1+x)^121)`
`d/(dx)(1/((1+x)^120))`

`=-120/((1+x)^121)*d/(dx)(1+x)`

`d/(dx)(1+x)=1`
`d/(dx)(1+x)`

`=d/(dx)(1)+d/(dx)(x)`

`=0+1`

`=1`


`=-120/((1+x)^121)*1`

`=-120/((1+x)^121)`


`=-94.7643+0-(-120/((1+x)^121))`

`=-94.7643+0+120/((1+x)^121)`

`=-94.7643+120/((1+x)^121)`


`:. f'(x) = -94.7643+120/((1+x)^121)`

`x_0 = 0.1`


`1^(st)` iteration :

`f(x_0)=f(0.1)=-94.7643*0.1+1-1/((1+0.1)^120)=-8.476441`

`f'(x_0)=f'(0.1)=-94.7643+120/((1+0.1)^121)=-94.763123`

`x_1 = x_0 - f(x_0)/(f'(x_0))`

`x_1=0.1 - (-8.476441)/(-94.763123)`

`x_1=0.010551`


`2^(nd)` iteration :

`f(x_1)=f(0.010551)=-94.7643*0.010551+1-1/((1+0.010551)^120)=-0.283674`

`f'(x_1)=f'(0.010551)=-94.7643+120/((1+0.010551)^121)=-61.06503`

`x_2 = x_1 - f(x_1)/(f'(x_1))`

`x_2=0.010551 - (-0.283674)/(-61.06503)`

`x_2=0.005906`


`3^(rd)` iteration :

`f(x_2)=f(0.005906)=-94.7643*0.005906+1-1/((1+0.005906)^120)=-0.052973`

`f'(x_2)=f'(0.005906)=-94.7643+120/((1+0.005906)^121)=-35.914489`

`x_3 = x_2 - f(x_2)/(f'(x_2))`

`x_3=0.005906 - (-0.052973)/(-35.914489)`

`x_3=0.004431`


`4^(th)` iteration :

`f(x_3)=f(0.004431)=-94.7643*0.004431+1-1/((1+0.004431)^120)=-0.008181`

`f'(x_3)=f'(0.004431)=-94.7643+120/((1+0.004431)^121)=-24.480398`

`x_4 = x_3 - f(x_3)/(f'(x_3))`

`x_4=0.004431 - (-0.008181)/(-24.480398)`

`x_4=0.004097`


`5^(th)` iteration :

`f(x_4)=f(0.004097)=-94.7643*0.004097+1-1/((1+0.004097)^120)=-0.000479`

`f'(x_4)=f'(0.004097)=-94.7643+120/((1+0.004097)^121)=-21.592676`

`x_5 = x_4 - f(x_4)/(f'(x_4))`

`x_5=0.004097 - (-0.000479)/(-21.592676)`

`x_5=0.004074`


Approximate root of the equation `-94.7643x+1-1/((1+x)^120)=0` using Newton Raphson method is `0.004074` (After 5 iterations)

`n``x_0``f(x_0)``f'(x_0)``x_1`Update
10.1-8.476441-94.7631230.010551`x_0 = x_1`
20.010551-0.283674-61.065030.005906`x_0 = x_1`
30.005906-0.052973-35.9144890.004431`x_0 = x_1`
40.004431-0.008181-24.4803980.004097`x_0 = x_1`
50.004097-0.000479-21.5926760.004074`x_0 = x_1`



`=>i=0.004074`

Annual rate `=i*12`

Annual rate `=0.004074*12`

Annual rate `=0.048894`

Annual rate `=4.8894%`

Create Amortization Schedule
Month = 1
`12000*0.004074=48.8937`

`126.63-48.8937=77.7363`

`12000-77.7363=11922.2637`

Month = 2
`11922.2637*0.004074=48.5769`

`126.63-48.5769=78.0531`

`11922.2637-78.0531=11844.2106`


Amortization table
MonthDatePrincipalInterestPaymentTotal Interest PaidTotal Remaining Balance
012000
1Mar-202477.736348.8937126.6348.893711922.2637
2Apr-202478.053148.5769126.6397.470611844.2106
3May-202478.371148.2589126.63145.729511765.8395
4Jun-202478.690447.9396126.63193.669111687.1491
5Jul-202479.01147.619126.63241.288111608.1381
6Aug-202479.33347.297126.63288.585111528.8051
7Sep-202479.656246.9738126.63335.558911449.1489
8Oct-202479.980846.6492126.63382.208111369.1681
9Nov-202480.306746.3233126.63428.531411288.8614
10Dec-202480.633945.9961126.63474.527511208.2275
11Jan-202580.962445.6676126.63520.195111127.2651
12Feb-202581.292345.3377126.63565.532811045.9728
13Mar-202581.623545.0065126.63610.539310964.3493
14Apr-202581.956144.6739126.63655.213210882.3932
15May-202582.2944.34126.63699.553210800.1032
16Jun-202582.625344.0047126.63743.557910717.4779
17Jul-202582.961943.6681126.63787.22610634.516
18Aug-202583.343.33126.63830.55610551.216
19Sep-202583.639442.9906126.63873.546610467.5766
20Oct-202583.980242.6498126.63916.196410383.5964
21Nov-202584.322342.3077126.63958.504110299.2741
22Dec-202584.665941.9641126.631000.468210214.6082
23Jan-202685.010941.6191126.631042.087310129.5973
24Feb-202685.357241.2728126.631083.360110044.2401
25Mar-202685.70540.925126.631124.28519958.5351
26Apr-202686.054240.5758126.631164.86099872.4809
27May-202686.404940.2251126.631205.0869786.076
28Jun-202686.756939.8731126.631244.95919699.3191
29Jul-202687.110439.5196126.631284.47879612.2087
30Aug-202687.465339.1647126.631323.64349524.7434
31Sep-202687.821738.8083126.631362.45179436.9217
32Oct-202688.179538.4505126.631400.90229348.7422
33Nov-202688.538838.0912126.631438.99349260.2034
34Dec-202688.899637.7304126.631476.72389171.3038
35Jan-202789.261837.3682126.631514.0929082.042
36Feb-202789.625537.0045126.631551.09658992.4165
37Mar-202789.990736.6393126.631587.73588902.4258
38Apr-202790.357336.2727126.631624.00858812.0685
39May-202790.725535.9045126.631659.9138721.343
40Jun-202791.095135.5349126.631695.44798630.2479
41Jul-202791.466335.1637126.631730.61168538.7816
42Aug-202791.83934.791126.631765.40268446.9426
43Sep-202792.213234.4168126.631799.81948354.7294
44Oct-202792.588934.0411126.631833.86058262.1405
45Nov-202792.966133.6639126.631867.52448169.1744
46Dec-202793.344933.2851126.631900.80958075.8295
47Jan-202893.725332.9047126.631933.71427982.1042
48Feb-202894.107132.5229126.631966.23717887.9971
49Mar-202894.490632.1394126.631998.37657793.5065
50Apr-202894.875631.7544126.632030.13097698.6309
51May-202895.262231.3678126.632061.49877603.3687
52Jun-202895.650330.9797126.632092.47847507.7184
53Jul-202896.0430.59126.632123.06847411.6784
54Aug-202896.431330.1987126.632153.26717315.2471
55Sep-202896.824229.8058126.632183.07297218.4229
56Oct-202897.218729.4113126.632212.48427121.2042
57Nov-202897.614929.0151126.632241.49937023.5893
58Dec-202898.012628.6174126.632270.11676925.5767
59Jan-202998.411928.2181126.632298.33486827.1648
60Feb-202998.812927.8171126.632326.15196728.3519
61Mar-202999.215527.4145126.632353.56646629.1364
62Apr-202999.619827.0102126.632380.57666529.5166
63May-2029100.025726.6043126.632407.18096429.4909
64Jun-2029100.433226.1968126.632433.37776329.0577
65Jul-2029100.842425.7876126.632459.16536228.2153
66Aug-2029101.253325.3767126.632484.5426126.962
67Sep-2029101.665924.9641126.632509.50616025.2961
68Oct-2029102.080124.5499126.632534.0565923.216
69Nov-2029102.49624.134126.632558.195820.72
70Dec-2029102.913623.7164126.632581.90645717.8064
71Jan-2030103.33323.297126.632605.20345614.4734
72Feb-2030103.75422.876126.632628.07945510.7194
73Mar-2030104.176722.4533126.632650.53275406.5427
74Apr-2030104.601222.0288126.632672.56155301.9415
75May-2030105.027421.6026126.632694.16415196.9141
76Jun-2030105.455321.1747126.632715.33885091.4588
77Jul-2030105.88520.745126.632736.08384985.5738
78Aug-2030106.316420.3136126.632756.39744879.2574
79Sep-2030106.749619.8804126.632776.27784772.5078
80Oct-2030107.184619.4454126.632795.72324665.3232
81Nov-2030107.621319.0087126.632814.73194557.7019
82Dec-2030108.059818.5702126.632833.30214449.6421
83Jan-2031108.500118.1299126.632851.4324341.142
84Feb-2031108.942117.6879126.632869.11994232.1999
85Mar-2031109.38617.244126.632886.36394122.8139
86Apr-2031109.831716.7983126.632903.16224012.9822
87May-2031110.279216.3508126.632919.5133902.703
88Jun-2031110.728515.9015126.632935.41453791.9745
89Jul-2031111.179715.4503126.632950.86483680.7948
90Aug-2031111.632714.9973126.632965.86213569.1621
91Sep-2031112.087614.5424126.632980.40453457.0745
92Oct-2031112.544214.0858126.632994.49033344.5303
93Nov-2031113.002813.6272126.633008.11753231.5275
94Dec-2031113.463213.1668126.633021.28433118.0643
95Jan-2032113.925512.7045126.633033.98883004.1388
96Feb-2032114.389712.2403126.633046.22912889.7491
97Mar-2032114.855811.7742126.633058.00332774.8933
98Apr-2032115.323811.3062126.633069.30952659.5695
99May-2032115.793710.8363126.633080.14582543.7758
100Jun-2032116.265510.3645126.633090.51032427.5103
101Jul-2032116.73929.8908126.633100.40112310.7711
102Aug-2032117.21489.4152126.633109.81632193.5563
103Sep-2032117.69248.9376126.633118.75392075.8639
104Oct-2032118.1728.458126.633127.21191957.6919
105Nov-2032118.65347.9766126.633135.18851839.0385
106Dec-2032119.13697.4931126.633142.68161719.9016
107Jan-2033119.62237.0077126.633149.68931600.2793
108Feb-2033120.10976.5203126.633156.20961480.1696
109Mar-2033120.59916.0309126.633162.24051359.5705
110Apr-2033121.09055.5395126.633167.781238.48
111May-2033121.58385.0462126.633172.82621116.8962
112Jun-2033122.07924.5508126.633177.377994.817
113Jul-2033122.57664.0534126.633181.4304872.2404
114Aug-2033123.07613.5539126.633184.9843749.1643
115Sep-2033123.57763.0524126.633188.0367625.5867
116Oct-2033124.08112.5489126.633190.5856501.5056
117Nov-2033124.58662.0434126.633192.629376.919
118Dec-2033125.09431.5357126.633194.1647251.8247
119Jan-2034125.60391.0261126.633195.1908126.2208
120Feb-2034126.22080.5143126.73513195.70510
2. Find Interest Rate i = ?
for Monthly payment A = 2379, Loan amount P = 100000, Loan term n = 5 years,

Solution:
`P=100000` (Loan amount)

`n=5` years (Number of periods)

Monthly Amount `A=2379` per month

`n=5*12=60` months

`A=(P*i)/(1-(1+i)^-n)`

`:.2379=(100000*i)/(1-(1+i)^-60)`

`:.(1-(1+i)^-60)/(i)=(100000)/(2379)`

`:.(1-(1+i)^-60)/(i)=42.0345`

Now, find one solution using Newton Raphson method

Here `1-(1+x)^-60=42.0345x`

`:.-42.0345x+1-1/((1+x)^60)=0`

Let `f(x) = -42.0345x+1-1/((1+x)^60)`

`d/(dx)(-42.0345x+1-1/((1+x)^60))=-42.0345+60/((1+x)^61)`


`d/(dx)(-42.0345x+1-1/((1+x)^60))`

`=-d/(dx)(42.0345x)+d/(dx)(1)-d/(dx)(1/((1+x)^60))`

`d/(dx)(1/((1+x)^60))=-60/((1+x)^61)`
`d/(dx)(1/((1+x)^60))`

`=-60/((1+x)^61)*d/(dx)(1+x)`

`d/(dx)(1+x)=1`
`d/(dx)(1+x)`

`=d/(dx)(1)+d/(dx)(x)`

`=0+1`

`=1`


`=-60/((1+x)^61)*1`

`=-60/((1+x)^61)`


`=-42.0345+0-(-60/((1+x)^61))`

`=-42.0345+0+60/((1+x)^61)`

`=-42.0345+60/((1+x)^61)`


`:. f'(x) = -42.0345+60/((1+x)^61)`

`x_0 = 0.1`


`1^(st)` iteration :

`f(x_0)=f(0.1)=-42.0345*0.1+1-1/((1+0.1)^60)=-3.206734`

`f'(x_0)=f'(0.1)=-42.0345+60/((1+0.1)^61)=-41.855358`

`x_1 = x_0 - f(x_0)/(f'(x_0))`

`x_1=0.1 - (-3.206734)/(-41.855358)`

`x_1=0.023385`


`2^(nd)` iteration :

`f(x_1)=f(0.023385)=-42.0345*0.023385+1-1/((1+0.023385)^60)=-0.232823`

`f'(x_1)=f'(0.023385)=-42.0345+60/((1+0.023385)^61)=-27.387094`

`x_2 = x_1 - f(x_1)/(f'(x_1))`

`x_2=0.023385 - (-0.232823)/(-27.387094)`

`x_2=0.014884`


`3^(rd)` iteration :

`f(x_2)=f(0.014884)=-42.0345*0.014884+1-1/((1+0.014884)^60)=-0.037756`

`f'(x_2)=f'(0.014884)=-42.0345+60/((1+0.014884)^61)=-17.670589`

`x_3 = x_2 - f(x_2)/(f'(x_2))`

`x_3=0.014884 - (-0.037756)/(-17.670589)`

`x_3=0.012747`


`4^(th)` iteration :

`f(x_3)=f(0.012747)=-42.0345*0.012747+1-1/((1+0.012747)^60)=-0.003493`

`f'(x_3)=f'(0.012747)=-42.0345+60/((1+0.012747)^61)=-14.32809`

`x_4 = x_3 - f(x_3)/(f'(x_3))`

`x_4=0.012747 - (-0.003493)/(-14.32809)`

`x_4=0.012504`


`5^(th)` iteration :

`f(x_4)=f(0.012504)=-42.0345*0.012504+1-1/((1+0.012504)^60)=-0.00005`

`f'(x_4)=f'(0.012504)=-42.0345+60/((1+0.012504)^61)=-13.918183`

`x_5 = x_4 - f(x_4)/(f'(x_4))`

`x_5=0.012504 - (-0.00005)/(-13.918183)`

`x_5=0.0125`


Approximate root of the equation `-42.0345x+1-1/((1+x)^60)=0` using Newton Raphson method is `0.0125` (After 5 iterations)

`n``x_0``f(x_0)``f'(x_0)``x_1`Update
10.1-3.206734-41.8553580.023385`x_0 = x_1`
20.023385-0.232823-27.3870940.014884`x_0 = x_1`
30.014884-0.037756-17.6705890.012747`x_0 = x_1`
40.012747-0.003493-14.328090.012504`x_0 = x_1`
50.012504-0.00005-13.9181830.0125`x_0 = x_1`



`=>i=0.0125`

Annual rate `=i*12`

Annual rate `=0.0125*12`

Annual rate `=0.150001`

Annual rate `=15.0001%`

Create Amortization Schedule
Month = 1
`100000*0.0125=1250.0083`

`2379-1250.0083=1128.9917`

`100000-1128.9917=98871.0083`

Month = 2
`98871.0083*0.0125=1235.8958`

`2379-1235.8958=1143.1042`

`98871.0083-1143.1042=97727.9041`


Amortization table
MonthDatePrincipalInterestPaymentTotal Interest PaidTotal Remaining Balance
0100000
1Mar-20241128.99171250.008323791250.008398871.0083
2Apr-20241143.10421235.895823792485.904197727.9041
3May-20241157.39311221.606923793707.51196570.511
4Jun-20241171.86061207.139423794914.650495398.6504
5Jul-20241186.50891192.491123796107.141594212.1415
6Aug-20241201.34041177.659623797284.801193010.8011
7Sep-20241216.35721162.642823798447.443991794.4439
8Oct-20241231.56181147.438223799594.882190562.8821
9Nov-20241246.95641132.0436237910726.925789315.9257
10Dec-20241262.54351116.4565237911843.382288053.3822
11Jan-20251278.32541100.6746237912944.056886775.0568
12Feb-20251294.30461084.6954237914028.752285480.7522
13Mar-20251310.48351068.5165237915097.268784170.2687
14Apr-20251326.86461052.1354237916149.404182843.4041
15May-20251343.45061035.5494237917184.953581499.9535
16Jun-20251360.24381018.7562237918203.709780139.7097
17Jul-20251377.2471001.753237919205.462778762.4627
18Aug-20251394.4627984.537323792019077368
19Sep-20251411.8936967.1064237921157.106475956.1064
20Oct-20251429.5423949.4577237922106.564174526.5641
21Nov-20251447.4117931.5883237923038.152473079.1524
22Dec-20251465.5045913.4955237923951.647971613.6479
23Jan-20261483.8234895.1766237924846.824570129.8245
24Feb-20261502.3714876.6286237925723.453168627.4531
25Mar-20261521.1511857.8489237926581.30267106.302
26Apr-20261540.1656838.8344237927420.136465566.1364
27May-20261559.4178819.5822237928239.718664006.7186
28Jun-20261578.9107800.0893237929039.807962427.8079
29Jul-20261598.6472780.3528237929820.160760829.1607
30Aug-20261618.6304760.3696237930580.530359210.5303
31Sep-20261638.8634740.1366237931320.666957571.6669
32Oct-20261659.3494719.6506237932040.317555912.3175
33Nov-20261680.0914698.9086237932739.226154232.2261
34Dec-20261701.0927677.9073237933417.133452531.1334
35Jan-20271722.3565656.6435237934073.776950808.7769
36Feb-20271743.8861635.1139237934708.890849064.8908
37Mar-20271765.6848613.3152237935322.20647299.206
38Apr-20271787.756591.244237935913.4545511.45
39May-20271810.1031568.8969237936482.346943701.3469
40Jun-20271832.7295546.2705237937028.617441868.6174
41Jul-20271855.6388523.3612237937551.978640012.9786
42Aug-20271878.8344500.1656237938052.144238134.1442
43Sep-20271902.32476.68237938528.824236231.8242
44Oct-20271926.0992452.9008237938981.72534305.725
45Nov-20271950.1756428.8244237939410.549432355.5494
46Dec-20271974.5529404.4471237939814.996530380.9965
47Jan-20281999.235379.765237940194.761528381.7615
48Feb-20282024.2256354.7744237940549.535926357.5359
49Mar-20282049.5286329.4714237940879.007324308.0073
50Apr-20282075.1479303.8521237941182.859422232.8594
51May-20282101.0874277.9126237941460.77220131.772
52Jun-20282127.3512251.6488237941712.420818004.4208
53Jul-20282153.9432225.0568237941937.477615850.4776
54Aug-20282180.8677198.1323237942135.609913669.6099
55Sep-20282208.1287170.8713237942306.481211461.4812
56Oct-20282235.7305143.2695237942449.75079225.7507
57Nov-20282263.6773115.3227237942565.07346962.0734
58Dec-20282291.973587.0265237942652.09994670.0999
59Jan-20292320.623458.3766237942710.47652349.4765
60Feb-20292349.476529.36872378.845242739.84520
3. Find Interest Rate i = ?
for Monthly payment A = 507.25, Loan amount P = 20000, Loan term n = 4 years,

Solution:
`P=20000` (Loan amount)

`n=4` years (Number of periods)

Monthly Amount `A=507.25` per month

`n=4*12=48` months

`A=(P*i)/(1-(1+i)^-n)`

`:.507.25=(20000*i)/(1-(1+i)^-48)`

`:.(1-(1+i)^-48)/(i)=(20000)/(507.25)`

`:.(1-(1+i)^-48)/(i)=39.4283`

Now, find one solution using Newton Raphson method

Here `1-(1+x)^-48=39.4283x`

`:.-39.4283x+1-1/((1+x)^48)=0`

Let `f(x) = -39.4283x+1-1/((1+x)^48)`

`d/(dx)(-39.4283x+1-1/((1+x)^48))=-39.4283+48/((1+x)^49)`


`d/(dx)(-39.4283x+1-1/((1+x)^48))`

`=-d/(dx)(39.4283x)+d/(dx)(1)-d/(dx)(1/((1+x)^48))`

`d/(dx)(1/((1+x)^48))=-48/((1+x)^49)`
`d/(dx)(1/((1+x)^48))`

`=-48/((1+x)^49)*d/(dx)(1+x)`

`d/(dx)(1+x)=1`
`d/(dx)(1+x)`

`=d/(dx)(1)+d/(dx)(x)`

`=0+1`

`=1`


`=-48/((1+x)^49)*1`

`=-48/((1+x)^49)`


`=-39.4283+0-(-48/((1+x)^49))`

`=-39.4283+0+48/((1+x)^49)`

`=-39.4283+48/((1+x)^49)`


`:. f'(x) = -39.4283+48/((1+x)^49)`

`x_0 = 0.1`


`1^(st)` iteration :

`f(x_0)=f(0.1)=-39.4283*0.1+1-1/((1+0.1)^48)=-2.953137`

`f'(x_0)=f'(0.1)=-39.4283+48/((1+0.1)^49)=-38.97852`

`x_1 = x_0 - f(x_0)/(f'(x_0))`

`x_1=0.1 - (-2.953137)/(-38.97852)`

`x_1=0.024237`


`2^(nd)` iteration :

`f(x_1)=f(0.024237)=-39.4283*0.024237+1-1/((1+0.024237)^48)=-0.272413`

`f'(x_1)=f'(0.024237)=-39.4283+48/((1+0.024237)^49)=-24.581843`

`x_2 = x_1 - f(x_1)/(f'(x_1))`

`x_2=0.024237 - (-0.272413)/(-24.581843)`

`x_2=0.013155`


`3^(rd)` iteration :

`f(x_2)=f(0.013155)=-39.4283*0.013155+1-1/((1+0.013155)^48)=-0.052697`

`f'(x_2)=f'(0.013155)=-39.4283+48/((1+0.013155)^49)=-14.128063`

`x_3 = x_2 - f(x_2)/(f'(x_2))`

`x_3=0.013155 - (-0.052697)/(-14.128063)`

`x_3=0.009425`


`4^(th)` iteration :

`f(x_3)=f(0.009425)=-39.4283*0.009425+1-1/((1+0.009425)^48)=-0.00906`

`f'(x_3)=f'(0.009425)=-39.4283+48/((1+0.009425)^49)=-9.116324`

`x_4 = x_3 - f(x_3)/(f'(x_3))`

`x_4=0.009425 - (-0.00906)/(-9.116324)`

`x_4=0.008431`


`5^(th)` iteration :

`f(x_4)=f(0.008431)=-39.4283*0.008431+1-1/((1+0.008431)^48)=-0.000739`

`f'(x_4)=f'(0.008431)=-39.4283+48/((1+0.008431)^49)=-7.617439`

`x_5 = x_4 - f(x_4)/(f'(x_4))`

`x_5=0.008431 - (-0.000739)/(-7.617439)`

`x_5=0.008334`


`6^(th)` iteration :

`f(x_5)=f(0.008334)=-39.4283*0.008334+1-1/((1+0.008334)^48)=-0.000007`

`f'(x_5)=f'(0.008334)=-39.4283+48/((1+0.008334)^49)=-7.467189`

`x_6 = x_5 - f(x_5)/(f'(x_5))`

`x_6=0.008334 - (-0.000007)/(-7.467189)`

`x_6=0.008333`


Approximate root of the equation `-39.4283x+1-1/((1+x)^48)=0` using Newton Raphson method is `0.008333` (After 6 iterations)

`n``x_0``f(x_0)``f'(x_0)``x_1`Update
10.1-2.953137-38.978520.024237`x_0 = x_1`
20.024237-0.272413-24.5818430.013155`x_0 = x_1`
30.013155-0.052697-14.1280630.009425`x_0 = x_1`
40.009425-0.00906-9.1163240.008431`x_0 = x_1`
50.008431-0.000739-7.6174390.008334`x_0 = x_1`
60.008334-0.000007-7.4671890.008333`x_0 = x_1`



`=>i=0.008333`

Annual rate `=i*12`

Annual rate `=0.008333*12`

Annual rate `=0.099998`

Annual rate `=9.9998%`

Create Amortization Schedule
Month = 1
`20000*0.008333=166.6635`

`507.25-166.6635=340.5865`

`20000-340.5865=19659.4135`

Month = 2
`19659.4135*0.008333=163.8254`

`507.25-163.8254=343.4246`

`19659.4135-343.4246=19315.9889`


Amortization table
MonthDatePrincipalInterestPaymentTotal Interest PaidTotal Remaining Balance
020000
1Mar-2024340.5865166.6635507.25166.663519659.4135
2Apr-2024343.4246163.8254507.25330.488919315.9889
3May-2024346.2864160.9636507.25491.452518969.7025
4Jun-2024349.1721158.0779507.25649.530418620.5304
5Jul-2024352.0818155.1682507.25804.698618268.4486
6Aug-2024355.0158152.2342507.25956.932817913.4328
7Sep-2024357.9742149.2758507.251106.208617555.4586
8Oct-2024360.9573146.2927507.251252.501317194.5013
9Nov-2024363.9652143.2848507.251395.786116830.5361
10Dec-2024366.9982140.2518507.251536.037916463.5379
11Jan-2025370.0564137.1936507.251673.231516093.4815
12Feb-2025373.1402134.1098507.251807.341315720.3413
13Mar-2025376.2496131.0004507.251938.341715344.0917
14Apr-2025379.385127.865507.252066.206714964.7067
15May-2025382.5464124.7036507.252190.910314582.1603
16Jun-2025385.7343121.5157507.252312.42614196.426
17Jul-2025388.9487118.3013507.252430.727313807.4773
18Aug-2025392.1898115.0602507.252545.787513415.2875
19Sep-2025395.458111.792507.252657.579513019.8295
20Oct-2025398.7535108.4965507.252766.07612621.076
21Nov-2025402.0763105.1737507.252871.249712218.9997
22Dec-2025405.4269101.8231507.252973.072811813.5728
23Jan-2026408.805498.4446507.253071.517411404.7674
24Feb-2026412.212195.0379507.253166.555310992.5553
25Mar-2026415.647191.6029507.253258.158210576.9082
26Apr-2026419.110788.1393507.253346.297510157.7975
27May-2026422.603384.6467507.253430.94429735.1942
28Jun-2026426.124981.1251507.253512.06939309.0693
29Jul-2026429.675977.5741507.253589.64348879.3934
30Aug-2026433.256473.9936507.253663.6378446.137
31Sep-2026436.866870.3832507.253734.02028009.2702
32Oct-2026440.507366.7427507.253800.76297568.7629
33Nov-2026444.178263.0718507.253863.83477124.5847
34Dec-2026447.879659.3704507.253923.20516676.7051
35Jan-2027451.611855.6382507.253978.84336225.0933
36Feb-2027455.375251.8748507.254030.71815769.7181
37Mar-2027459.169948.0801507.254078.79825310.5482
38Apr-2027462.996344.2537507.254123.05194847.5519
39May-2027466.854540.3955507.254163.44744380.6974
40Jun-2027470.744936.5051507.254199.95253909.9525
41Jul-2027474.667732.5823507.254232.53483435.2848
42Aug-2027478.623228.6268507.254261.16162956.6616
43Sep-2027482.611624.6384507.254285.82474.05
44Oct-2027486.633320.6167507.254306.41671987.4167
45Nov-2027490.688516.5615507.254322.97821496.7282
46Dec-2027494.777512.4725507.254335.45071001.9507
47Jan-2028498.90068.3494507.254343.8001503.0501
48Feb-2028503.05014.192507.24214347.99210



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2. Example : Loan amount (P)
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