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How Fixed-Rate Loan Amortization Works
or

Amortization Math:
How to construct a mortgage amortization table using formulas rather than mortgage calculators.

Mortgage Library: Ask a Mortgage Related Question: Answers: Amortization

Question:

Hello,

I used your mortgage calculator and entered in a loan of $100,000 at a 7% fixed interest for 30 years. It gave me a number of $139,508.90 after 30 years, that is the total interest paid over the life of the loan. This number does not make sense because 7% of $100,000 is only $7,000 and not $139,508.90. $139,508.90 is almost 140% of the original $100,000. Please explain your reasoning for this.

I look forward in hearing from you.

F.

Answer:

Dear F.,

If you borrowed $100,000 from a lender with an agreement that at the end of 30 years you would repay the original loan amount plus 7%, then your total repayment would be $107,000. This is not how mortgage loans work, as mortgages utilize a nominal interest rate: the interest rate per year. The repayment process is far more complicated, and involves the concept of amortization.

When money is loaned for 30 years, the mortgage agreement requires the borrower to make 360 periodic (monthly) payments to the lender. The payments must remain the same each month and fully repay both the interest and principal during the life of the loan.

The quoted interest rate of 7.00% per year is compounded 12 times a year, resulting in a monthly rate of 0.58% (which is computed by dividing the note rate by 12).

To calculate the interest due for a given month, the monthly rate is multiplied by the current loan balance. If you borrowed $100,000 at 7%, at the end of the first month your interest due would be $100,000 x (0.07 / 12).

The process of recalculating the interest and principal every month is called amortization.

To illustrate how each monthly payment is applied to your loan we need to construct an amortization table, reflecting each payment on it's own line. To build an actual amortization table on a 30-year fixed $100,000.00 loan at 7.00% we need to answer the following two questions:

1. How to compute the monthly payment for this loan?

2. How is the interest part calculated each month?

As soon as these questions are answered, the remaining part of each payment that goes monthly toward your loan balance is easily calculated by subtracting the interest part from the monthly payment.

We will derive the equation for the monthly payment PMT in the following manner:

Monthly Payment Calculation

Let the original loan balance be: LB(0) {it is the $100,000.00 you borrowed}

Then the interest due ID(1) at the end of the first payment period is:

ID(1) = LB(0) * i

where i is the effective interest rate per payment period.

Our payment period is 1 month (1 year/12), so the effective interest rate is 1/12 of the note rate. In our example, it is 0.07/12.

The principal part PP(1) of the first monthly payment (the part that goes toward the loan balance) is calculated by subtracting the interest part from the monthly payment:

PP(1) = PMT - ID(1)

The loan balance after the first payment LB(1) is calculated by subtracting the principal part (it was calculated above) from the original loan balance.

LB(1) = LB(0) - PP(1)
LB(1) = LB(0) - (PMT - ID(1))
LB(1) = LB(0) - (PMT - LB(0) * i)
LB(1) = LB(0)*(1 + i) - PMT

Similarly, the loan balance after the second payment is:

LB(2) = LB(1)*(1 + i) - PMT

Replacing the LB(1) with the expression we just derived, we get:

LB(2) = (LB(0)*(1 + i) - PMT)*(1 + i) - PMT

or

LB(2) = LB(0)*(1 + i)^2 - PMT*((1 + i) + 1)

The loan balance after the third payment is:

LB(3) = LB(0)*(1 + i)^3 - PMT*((1 + i)^2 + (1 + i) + 1)

The loan balance after n payments is:

LB(n) = LB(0)*(1 + i)^n - PMT*((1 + i)^(n-1) + ... + (1 + i) + 1)

The sum of the finite series: 1 + a + (a^2) + (a^3) + ... + (a^n) is (1-a^(n+1))/(1-a)

As it can be easily seen:
if SUM(n) is the series sum, then SUM(n) - a * SUM(n) = 1 - a^(n+1)
solving for SUM(n):
SUM(n) = (1-a^(n+1))/(1-a)

Now, with a simple re-arrangement, our equation for loan balance after n payments becomes:

LB(n) = LB(0)*(1 + i)^n - PMT*(1-(1 + i)^n)/(1-(1 + i))
LB(n) = LB(0)*(1 + i)^n - PMT*((1 + i)^n-1)/i

The loan balance after 360 payments is: $0.00 (the loan is paid off). So:

LB(0)*(1 + i)^360 = PMT*((1 + i)^360-1)/i

Here is the solution for PMT:

PMT = i * LB(0)*(1 + i)^360 / ((1 + i)^360-1)

Your monthly payment will be:

PMT = (0.07/12) * 100000*(1 + 0.07/12)^360 / ((1 + 0.07/12)^360-1) = 665.302495179183

It must be rounded to the nearest cent: 665.30.

Now, determining the interest part of each monthly payment is simple. It is computed by multiplying the current loan balance by the effective interest rate per payment period.

Your initial loan balance is $100,000.00.

When your first monthly payment is due, the interest part is: $100,000.00 * 0.07/12 = $583.33.

Let's calculate the remaining part of the payment (the principal part), that goes toward the amount borrowed:

$665.30 - $583.33 = $81.97.

When your second monthly payment is due, your loan balance is: $100,000.00 - $81.97 = $99,918.03.

The interest part is: $99,918.03 * 0.07/12 = $582.86.

The remaining part of the payment (the principal part) is: $665.30 - $582.86 = $82.45.

The total interest paid is: $583.33 + $582.86 = $1,166.19.

Now we have the first two rows of the loan amortization table:

 Amortization Table

Event Payment Interest Principal Total Interest Loan Balance
Loan         $100,000.00
Payment # 1 $665.30 $583.33 $81.97 $583.33 $99,918.03
Payment # 2 $665.30 $582.86 $82.45 $1,166.19 $99,835.58

The process repeats each month...
 
Payment # 3 $665.30 $582.37 $82.93 $1,748.56 $99,752.66
Payment # 4 $665.30 $581.89 $83.41 $2,330.45 $99,669.24
Payment # 5 $665.30 $581.40 $83.90 $2,911.86 $99,585.34

...
 
Payment # 359 $665.30 $7.69 $657.61 $139,505.04 $661.44
Payment # 360 $665.30 $3.86 $661.44 $139,508.90 $0.00

How to compute the total interest paid over the life of the loan:

How much did you borrow?

$100.000.

How much did you pay?

360 monthly payments, $665.30 each time.

You paid: 360 * $665.30 = $239,508.

The total interest paid over the life of the loan = $239,508 - $100.000 = $139,508.

If you paid exactly 665.3024952, the total interest paid over the life of the loan would be = $139,508.90.

We hope this is of assistance to you.

Best regards,

The Mortgage-X Team


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