Amortization is the process of repaying a loan in equal, monthly payments. This calculator lets you estimate your monthly loan repayments. The calculator will produce a full explanation of how the computation was carried out.
problem
If a mortgage is amortized over 10 yearss at an interest rate of 7% and monthly payments of \$25.3, what is the original value of the mortgage?
solution
Original value of the mortgage is \$ 245.39.
Summary
Principal borrowed: | $245.39 |
Annual Interest Rate: | 7% |
Total Payments: | 10 |
Monthly Payment amount: | $25.3 |
Total Interest Paid: | $7.61 |
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 0.07 so
$$ \begin{aligned} \text{Monthly Rate} &= \left( 1 + 0.07 \right)^{\Large{\frac{1}{12}}} - 1 \\ \text{Monthly Rate} &\approx 0.0056541 \end{aligned} $$NOTE: One of the most common mistakes is to simply divide annual rate by 12 to get monthly rate.
Step 2: Determine principal borrowed 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
$$ A = \$25.3 ~,~ i = 0.0057 ~~ \text{and} ~~ n = 10 $$After plugging the given information we have
$$ \begin{aligned} A &= \frac{P \cdot i}{1- (1+i)^{-n} } \\ \\ 25.3 &= \frac{ P \cdot 0.0057 }{ 1- ( 1+ 0.0057 )^{\large{-10}} } \\ \\ 25.3 &= \frac{ P \cdot 0.0057 }{ 1- ( 1.0057 )^{\large{-10}} } \\ \\ 25.3 &= \frac{ P \cdot 0.0057 }{ 1- 0.9447 } \\ \\ 25.3 &= \frac{ P \cdot 0.0057 }{ 1- 0.9447 } \\ \\ 25.3 &= \frac{ P \cdot 0.0057 }{ 0.0553 } \\ \\ 25.3 &= P \cdot 0.1031 \\ \\ P & \approx 245.39 \end{aligned} $$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 ( \$245.39 ) by monthly interest rate (0.0057).
$$ 245.39 \cdot 0.0057 = \color{red}{ 1.398723 }$$Subtract the interest from the first payment to see how much principal is paid with the first payment.
$$ 25.3 - 1.398723 = \color{blue}{ 23.901277 }$$Determine the new balance by subtract above result from the old balacnce.
$$ \text{new balance} = 245.39 - 23.901277 = \color{green}{ 221.488723 }$$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 | 245.39 | |||
1 | 25.3 | 23.9 | 1.4 | 221.49 |
2 | 25.3 | 24.04 | 1.26 | 197.45 |
3 | 25.3 | 24.17 | 1.13 | 173.28 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
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