Amortization calculator

problem

You want to take out a mortgage for \$50000 with monthly payments at 4.5%, and you can afford \$550 per month payments. How long would you have to make payments to pay off the mortgage.

solution

You need to made 111 monthly payments. ( 111 months = 9 years and 3 months )

Summary

Principal borrowed:	$50000
Annual Interest Rate:	4.5%
Total Payments:	111
Monthly Payment amount:	$550
Total Interest Paid:	$11050

Show Amortization Table

Month	Payment Required	Principal Paid	Interest Payment	Remaining Balance
0				50000
1	550	366.26	183.74	49633.74
2	550	367.61	182.39	49266.13
3	550	368.96	181.04	48897.17
4	550	370.31	179.69	48526.86
5	550	371.67	178.33	48155.19
6	550	373.04	176.96	47782.15
7	550	374.41	175.59	47407.74
8	550	375.79	174.21	47031.95
9	550	377.17	172.83	46654.78
10	550	378.55	171.45	46276.23
11	550	379.94	170.06	45896.29
12	550	381.34	168.66	45514.95
13	550	382.74	167.26	45132.21
14	550	384.15	165.85	44748.06
15	550	385.56	164.44	44362.5
16	550	386.98	163.02	43975.52
17	550	388.4	161.6	43587.12
18	550	389.83	160.17	43197.29
19	550	391.26	158.74	42806.03
20	550	392.7	157.3	42413.33
21	550	394.14	155.86	42019.19
22	550	395.59	154.41	41623.6
23	550	397.04	152.96	41226.56
24	550	398.5	151.5	40828.06
25	550	399.97	150.03	40428.09
26	550	401.43	148.57	40026.66
27	550	402.91	147.09	39623.75
28	550	404.39	145.61	39219.36
29	550	405.88	144.12	38813.48
30	550	407.37	142.63	38406.11
31	550	408.87	141.13	37997.24
32	550	410.37	139.63	37586.87
33	550	411.88	138.12	37174.99
34	550	413.39	136.61	36761.6
35	550	414.91	135.09	36346.69
36	550	416.43	133.57	35930.26
37	550	417.96	132.04	35512.3
38	550	419.5	130.5	35092.8
39	550	421.04	128.96	34671.76
40	550	422.59	127.41	34249.17
41	550	424.14	125.86	33825.03
42	550	425.7	124.3	33399.33
43	550	427.26	122.74	32972.07
44	550	428.83	121.17	32543.24
45	550	430.41	119.59	32112.83
46	550	431.99	118.01	31680.84
47	550	433.58	116.42	31247.26
48	550	435.17	114.83	30812.09
49	550	436.77	113.23	30375.32
50	550	438.38	111.62	29936.94
51	550	439.99	110.01	29496.95
52	550	441.6	108.4	29055.35
53	550	443.23	106.77	28612.12
54	550	444.86	105.14	28167.26
55	550	446.49	103.51	27720.77
56	550	448.13	101.87	27272.64
57	550	449.78	100.22	26822.86
58	550	451.43	98.57	26371.43
59	550	453.09	96.91	25918.34
60	550	454.76	95.24	25463.58
61	550	456.43	93.57	25007.15
62	550	458.1	91.9	24549.05
63	550	459.79	90.21	24089.26
64	550	461.48	88.52	23627.78
65	550	463.17	86.83	23164.61
66	550	464.87	85.13	22699.74
67	550	466.58	83.42	22233.16
68	550	468.3	81.7	21764.86
69	550	470.02	79.98	21294.84
70	550	471.75	78.25	20823.09
71	550	473.48	76.52	20349.61
72	550	475.22	74.78	19874.39
73	550	476.97	73.03	19397.42
74	550	478.72	71.28	18918.7
75	550	480.48	69.52	18438.22
76	550	482.24	67.76	17955.98
77	550	484.02	65.98	17471.96
78	550	485.79	64.21	16986.17
79	550	487.58	62.42	16498.59
80	550	489.37	60.63	16009.22
81	550	491.17	58.83	15518.05
82	550	492.97	57.03	15025.08
83	550	494.79	55.21	14530.29
84	550	496.6	53.4	14033.69
85	550	498.43	51.57	13535.26
86	550	500.26	49.74	13035
87	550	502.1	47.9	12532.9
88	550	503.94	46.06	12028.96
89	550	505.8	44.2	11523.16
90	550	507.65	42.35	11015.51
91	550	509.52	40.48	10505.99
92	550	511.39	38.61	9994.6
93	550	513.27	36.73	9481.33
94	550	515.16	34.84	8966.17
95	550	517.05	32.95	8449.12
96	550	518.95	31.05	7930.17
97	550	520.86	29.14	7409.31
98	550	522.77	27.23	6886.54
99	550	524.69	25.31	6361.85
100	550	526.62	23.38	5835.23
101	550	528.56	21.44	5306.67
102	550	530.5	19.5	4776.17
103	550	532.45	17.55	4243.72
104	550	534.41	15.59	3709.31
105	550	536.37	13.63	3172.94
106	550	538.34	11.66	2634.6
107	550	540.32	9.68	2094.28
108	550	542.3	7.7	1551.98
109	550	544.3	5.7	1007.68
110	550	546.3	3.7	461.38
111	463.08	461.38	1.7	0

Explanation

Step 1: Determine monthly interest rate.

The formula for changing from an annual interest rate to a monthly one is:

$$ \text{Monthly Rate} = \left( 1 + \text{annual rate}\right)^{\Large{\frac{1}{12}}} - 1 $$

In this example annual rate is 0.045 so

$$ \begin{aligned} \text{Monthly Rate} &= \left( 1 + 0.045 \right)^{\Large{\frac{1}{12}}} - 1 \\ \text{Monthly Rate} &\approx 0.0036748 \end{aligned} $$

NOTE: One of the most common mistakes is to simply divide annual rate by 12 to get monthly rate.

Step 2: Determine monthly payment by using the following formula

$$ A = \frac{P \cdot i}{1- (1+i)^{-n} } $$

A = monthly payment amount P = loan amount i = monthly interest rate n = total number of payments

In this example we have

$$ A = $550 ~,~ P = 50000 ~~ \text{and} ~~ i = 0.0036748 $$

After plugging the given information we have

$$ \begin{aligned} A &= \frac{P \cdot i}{1- (1+i)^{-n} } \\ 1 - (1+i)^{-n} &= \frac{P \cdot i}{A} \\ 1 - (1+0.0036748)^{-n} &= \frac{ 183.74 }{ 550 } \\ 1 - ( 1.0036748 )^{-n} &= 0.33407272727273 \\ ( 1.0036748 )^{-n} &= 1 - 0.33407272727273 \\ ( 1.0036748 )^{-n} &= 0.66592727272727 \\ \ln( 1.0036748 )^{-n} &= ln( 0.66592727272727 ) \\ -n \cdot \ln( 1.0036748 ) &= ln( 0.66592727272727 ) \\ -n &= \frac{ -0.40657481451371 }{ 0.0036680644187127 } \\ n &= 111 ~~ \text{payments} \end{aligned} $$

Step 3: Create Amortization Schedule by finding the breakdown of each monthly payment.

The monthly interest to be paid in the first payment is calculated by multiply the remaining balance ( \$50000 ) by monthly interest rate (0.0036748).

$$ 50000 \cdot 0.0036748 = \color{red}{ 183.74 }$$

Subtract the interest from the first payment to see how much principal is paid with the first payment.

$$ 550 - 183.74 = \color{blue}{ 366.26 }$$

Determine the new balance by subtract above result from the old balacnce.

$$ \text{new balance} = 50000 - 366.26 = \color{green}{ 49633.74 }$$

Above steps can be repeated for each payment to construct the amortization schedule table. First 4 rows of the amortization table are:

Month	Payment Required	Principal Paid	Interest Payment	Remaining Balance
0				50000
1	550	366.26	183.74	49633.74
2	550	367.61	182.39	49266.13
3	550	368.96	181.04	48897.17
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