# Mortgage Monthly Payment Calculator

#### Mortgage Monthly Payment Definition:

Monthly payment calculator gives the amount to be paid monthly depending upon the loan amount taken, the duration of the loan and the interest rate.

The loan taken may be to purchase a car or a real estate. The borrower is liable to pay the amount monthly to the lender. In the case of a default the lender has every right on the property which the borrower took by the amount in debt to the lender.

#### Monthly payment formula:

The monthly payment for a fixed rate mortgage, depends on the monthly interest rate R, the number of monthly payments period P and the amount borrowed A then the monthly payment M is given by the formula

M = (R/1-(1+R) ^ -P)) A

A home loan for \$200,000 with a fixed yearly interest of 6.5% for 30 years , monthly payment is
M = (R/(1-(1+R) ^ -P)) A = \$200,000 (6.5/100/12 / 1- (1+6.5 /100/12) ^ -360))
= \$1264.14

In the above formula 360 = 30*12 i.e. the number of years are multiplied with the number of months in a year to get the monthly payment amount \$1264.14

#### Derivation of the formula:

The amount owed on the loan every month is nothing but the amount owed from the previous month plus the interest on the amount minus the fixed amount paid every month.
i.e.

• The amount owed on month 0: A
• Amount owed on month 1: (1+R) A – M
• Amount owed on month 2: (1+R) ((1+R) A –M) – M = (1+R)^2 A – (1+(1+R))M
• amount owed on month 3: (1+R)((1+R)((1+R) A – M) – M) – M= (1+R)^3 A – (1+(1+R)+(1+R)^2) M

and so on

This comes to the formula
The amount owed on month P
= (1+R) ^P * A – the sum of the terms polynomial (1+R) M
Which is nothing but (1+R) ^P *A – ((1+R) ^P -1/ (1+R)-1) * M
= (1+R) ^P *A – (1+R) ^P-1/R *M

Here M = R (1+R) ^P/(1+R)^P -1 *A
= R/ (1-(1+R) ^-P) *A

This monthly payment is chosen so that the amount is completely paid with full interest by the end of the term and no more money is owed.