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Amortization calculator

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

Show Amortization Table

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:

MonthPayment RequiredPrincipal PaidInterest PaymentRemaining Balance
0245.39
125.3 23.9 1.4 221.49
225.324.041.26197.45
325.324.171.13173.28

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Amortization calculator
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Examples
example 1:ex 1:
What is the monthly payment on a mortgage of $12,000 with an annual interest rate of 5.5% that runs for 10 years?
example 2:ex 2:
What is the original value of a mortgage if it is amortized over 10 months at 7% interest and $25.3 monthly payments?
example 3:ex 3:
You wish to take out a $50000 mortgage with monthly payments of 4.5%, and you can afford $550 per month. How long would it take for you to pay off the mortgage?
example 4:ex 4:
What is the interest rate on a $23,000 mortgage with $350 monthly payments for ten years?
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