problem
What is the monthly payment on a mortgage of \$12000 with
annual interest rate of 5.5% that runs for 10 years.
solution
The monthly payments is \$ 129.44.
Summary
Principal borrowed: | $12000 |
Annual Interest Rate: | 5.5% |
Total Payments: | 120 |
Monthly Payment amount: | $129.44 |
Total Interest Paid: | $3532.8 |
Show Amortization Table
Month | Payment Required | Principal Paid | Interest Payment | Remaining Balance |
0 | | | | 12000 |
1 | 129.44 | 75.78 | 53.66 | 11924.22 |
2 | 129.44 | 76.12 | 53.32 | 11848.1 |
3 | 129.44 | 76.46 | 52.98 | 11771.64 |
4 | 129.44 | 76.8 | 52.64 | 11694.84 |
5 | 129.44 | 77.14 | 52.3 | 11617.7 |
6 | 129.44 | 77.49 | 51.95 | 11540.21 |
7 | 129.44 | 77.84 | 51.6 | 11462.37 |
8 | 129.44 | 78.18 | 51.26 | 11384.19 |
9 | 129.44 | 78.53 | 50.91 | 11305.66 |
10 | 129.44 | 78.88 | 50.56 | 11226.78 |
11 | 129.44 | 79.24 | 50.2 | 11147.54 |
12 | 129.44 | 79.59 | 49.85 | 11067.95 |
13 | 129.44 | 79.95 | 49.49 | 10988 |
14 | 129.44 | 80.3 | 49.14 | 10907.7 |
15 | 129.44 | 80.66 | 48.78 | 10827.04 |
16 | 129.44 | 81.02 | 48.42 | 10746.02 |
17 | 129.44 | 81.39 | 48.05 | 10664.63 |
18 | 129.44 | 81.75 | 47.69 | 10582.88 |
19 | 129.44 | 82.12 | 47.32 | 10500.76 |
20 | 129.44 | 82.48 | 46.96 | 10418.28 |
21 | 129.44 | 82.85 | 46.59 | 10335.43 |
22 | 129.44 | 83.22 | 46.22 | 10252.21 |
23 | 129.44 | 83.6 | 45.84 | 10168.61 |
24 | 129.44 | 83.97 | 45.47 | 10084.64 |
25 | 129.44 | 84.34 | 45.1 | 10000.3 |
26 | 129.44 | 84.72 | 44.72 | 9915.58 |
27 | 129.44 | 85.1 | 44.34 | 9830.48 |
28 | 129.44 | 85.48 | 43.96 | 9745 |
29 | 129.44 | 85.86 | 43.58 | 9659.14 |
30 | 129.44 | 86.25 | 43.19 | 9572.89 |
31 | 129.44 | 86.63 | 42.81 | 9486.26 |
32 | 129.44 | 87.02 | 42.42 | 9399.24 |
33 | 129.44 | 87.41 | 42.03 | 9311.83 |
34 | 129.44 | 87.8 | 41.64 | 9224.03 |
35 | 129.44 | 88.19 | 41.25 | 9135.84 |
36 | 129.44 | 88.59 | 40.85 | 9047.25 |
37 | 129.44 | 88.98 | 40.46 | 8958.27 |
38 | 129.44 | 89.38 | 40.06 | 8868.89 |
39 | 129.44 | 89.78 | 39.66 | 8779.11 |
40 | 129.44 | 90.18 | 39.26 | 8688.93 |
41 | 129.44 | 90.59 | 38.85 | 8598.34 |
42 | 129.44 | 90.99 | 38.45 | 8507.35 |
43 | 129.44 | 91.4 | 38.04 | 8415.95 |
44 | 129.44 | 91.81 | 37.63 | 8324.14 |
45 | 129.44 | 92.22 | 37.22 | 8231.92 |
46 | 129.44 | 92.63 | 36.81 | 8139.29 |
47 | 129.44 | 93.04 | 36.4 | 8046.25 |
48 | 129.44 | 93.46 | 35.98 | 7952.79 |
49 | 129.44 | 93.88 | 35.56 | 7858.91 |
50 | 129.44 | 94.3 | 35.14 | 7764.61 |
51 | 129.44 | 94.72 | 34.72 | 7669.89 |
52 | 129.44 | 95.14 | 34.3 | 7574.75 |
53 | 129.44 | 95.57 | 33.87 | 7479.18 |
54 | 129.44 | 96 | 33.44 | 7383.18 |
55 | 129.44 | 96.42 | 33.02 | 7286.76 |
56 | 129.44 | 96.86 | 32.58 | 7189.9 |
57 | 129.44 | 97.29 | 32.15 | 7092.61 |
58 | 129.44 | 97.72 | 31.72 | 6994.89 |
59 | 129.44 | 98.16 | 31.28 | 6896.73 |
60 | 129.44 | 98.6 | 30.84 | 6798.13 |
61 | 129.44 | 99.04 | 30.4 | 6699.09 |
62 | 129.44 | 99.48 | 29.96 | 6599.61 |
63 | 129.44 | 99.93 | 29.51 | 6499.68 |
64 | 129.44 | 100.38 | 29.06 | 6399.3 |
65 | 129.44 | 100.82 | 28.62 | 6298.48 |
66 | 129.44 | 101.28 | 28.16 | 6197.2 |
67 | 129.44 | 101.73 | 27.71 | 6095.47 |
68 | 129.44 | 102.18 | 27.26 | 5993.29 |
69 | 129.44 | 102.64 | 26.8 | 5890.65 |
70 | 129.44 | 103.1 | 26.34 | 5787.55 |
71 | 129.44 | 103.56 | 25.88 | 5683.99 |
72 | 129.44 | 104.02 | 25.42 | 5579.97 |
73 | 129.44 | 104.49 | 24.95 | 5475.48 |
74 | 129.44 | 104.96 | 24.48 | 5370.52 |
75 | 129.44 | 105.42 | 24.02 | 5265.1 |
76 | 129.44 | 105.9 | 23.54 | 5159.2 |
77 | 129.44 | 106.37 | 23.07 | 5052.83 |
78 | 129.44 | 106.85 | 22.59 | 4945.98 |
79 | 129.44 | 107.32 | 22.12 | 4838.66 |
80 | 129.44 | 107.8 | 21.64 | 4730.86 |
81 | 129.44 | 108.29 | 21.15 | 4622.57 |
82 | 129.44 | 108.77 | 20.67 | 4513.8 |
83 | 129.44 | 109.26 | 20.18 | 4404.54 |
84 | 129.44 | 109.74 | 19.7 | 4294.8 |
85 | 129.44 | 110.23 | 19.21 | 4184.57 |
86 | 129.44 | 110.73 | 18.71 | 4073.84 |
87 | 129.44 | 111.22 | 18.22 | 3962.62 |
88 | 129.44 | 111.72 | 17.72 | 3850.9 |
89 | 129.44 | 112.22 | 17.22 | 3738.68 |
90 | 129.44 | 112.72 | 16.72 | 3625.96 |
91 | 129.44 | 113.23 | 16.21 | 3512.73 |
92 | 129.44 | 113.73 | 15.71 | 3399 |
93 | 129.44 | 114.24 | 15.2 | 3284.76 |
94 | 129.44 | 114.75 | 14.69 | 3170.01 |
95 | 129.44 | 115.26 | 14.18 | 3054.75 |
96 | 129.44 | 115.78 | 13.66 | 2938.97 |
97 | 129.44 | 116.3 | 13.14 | 2822.67 |
98 | 129.44 | 116.82 | 12.62 | 2705.85 |
99 | 129.44 | 117.34 | 12.1 | 2588.51 |
100 | 129.44 | 117.86 | 11.58 | 2470.65 |
101 | 129.44 | 118.39 | 11.05 | 2352.26 |
102 | 129.44 | 118.92 | 10.52 | 2233.34 |
103 | 129.44 | 119.45 | 9.99 | 2113.89 |
104 | 129.44 | 119.99 | 9.45 | 1993.9 |
105 | 129.44 | 120.52 | 8.92 | 1873.38 |
106 | 129.44 | 121.06 | 8.38 | 1752.32 |
107 | 129.44 | 121.6 | 7.84 | 1630.72 |
108 | 129.44 | 122.15 | 7.29 | 1508.57 |
109 | 129.44 | 122.69 | 6.75 | 1385.88 |
110 | 129.44 | 123.24 | 6.2 | 1262.64 |
111 | 129.44 | 123.79 | 5.65 | 1138.85 |
112 | 129.44 | 124.35 | 5.09 | 1014.5 |
113 | 129.44 | 124.9 | 4.54 | 889.6 |
114 | 129.44 | 125.46 | 3.98 | 764.14 |
115 | 129.44 | 126.02 | 3.42 | 638.12 |
116 | 129.44 | 126.59 | 2.85 | 511.53 |
117 | 129.44 | 127.15 | 2.29 | 384.38 |
118 | 129.44 | 127.72 | 1.72 | 256.66 |
119 | 129.44 | 128.29 | 1.15 | 128.37 |
120 | 128.94 | 128.37 | 0.57 | 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 0.055 so
$$ \begin{aligned}
\text{Monthly Rate} &= \left( 1 + 0.055 \right)^{\Large{\frac{1}{12}}} - 1 \\
\text{Monthly Rate} &\approx 0.0044717
\end{aligned}
$$
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
$$ P = $12000 ~,~ i = 0.0044717 ~~ \text{and} ~~ n = 12 \cdot 10 = 120 $$
After plugging the given information we have
$$ \begin{aligned}
A &= \frac{P \cdot i}{1- (1+i)^{-n} } \\
A &= \frac{ 12000 \cdot 0.0044717 }{ 1- ( 1+ 0.0044717 )^{\large{-120}} } \\
A &= \frac{ 53.6604 }{ 1 - ( 1.0044717 )^{\large{-120}} } \\
A &= 129.44
\end{aligned} $$
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 ( \$12000 ) by monthly interest rate (0.0044717).
$$ 12000 \cdot 0.0044717 = \color{red}{ 53.6604 }$$
Subtract the interest from the first payment to see how much principal
is paid with the first payment.
$$ 129.44 - 53.6604 = \color{blue}{ 75.7796 }$$
Determine the new balance by subtract above result from the old balacnce.
$$ \text{new balance} = 12000 - 75.7796 = \color{green}{ 11924.2204 }$$
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 | | | | 12000 |
1 | 129.44 | 75.78 | 53.66 | 11924.22 |
2 | 129.44 | 76.12 | 53.32 | 11848.1 |
3 | 129.44 | 76.46 | 52.98 | 11771.64 |
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