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- Calculate H = P*J, this is your current monthly interest
- Calculate C = M - H, this is your monthly payment minus your monthly interest, so it is the amount of principal you pay for the month.
- Calculate Q = P - C, this is the new balance of your principal of your loan.
- Set P = Q and repeat 1.

Definitions

N = No. of months of the mortgage payment M = Monthly mortgage payment J = Monthly interest rate P = Principal

For the first month N = 1 :

H = P*J C = M - P*J Q = P - (M - P*J) = P + PJ - M = P(1 + J) - M

For the second month N = 2 :

H = (P(1 + J) - M)*J C = M - [ PJ(1 + J) - MJ ] Q = P(1 + J) - M - (M - [ PJ(1 + J) - MJ ]) = P(1 + J) - M - M + PJ(1 + J) - MJ = P(1 + J)^{2}- M(1 + J) - M

For the third month N = 3 :

H = (P(1 + J)^{2}- M(1 + J) - M)*J C = M - [PJ(1 + J)^{2}- MJ(1 + J) - MJ] Q = P(1 + J)^{2}- M(1 + J) - M - (M - [PJ(1 + J)^{2}- MJ(1 + J) - MJ]) = P(1 + J)^{2}+ PJ(1 + J)^{2}- M(1 + J) - MJ(1 + J) - M - MJ - M = P(1 + J)^{3}- M(1 + J)^{2}- M(1 + J) - M[ Equation #1 ]

Let us digress and consider the Geometric series :

We know :

T _{n} = a r^{n} - 1

so the sum of the series is expressed as

S_{n} = a [ (1 - r^{n} ) / ( 1 - r ) ]

From [ Equation 1 ] we know that

**M(1 + J) ^{2} - M(1 + J) - M** is a Geometric series.

Where r is (1 + J) and a = M

Thus the sum of this series is equal to

S_{n}= M [ (1- (1 + J)^{n}) / (1- (1 + J)) ][ Equation #2 ]

Now substitute [ Equation 2 ] into [ Equation 1 ] and set Q = 0,

The reason why we set Q equal to zero is simple, when we finish paying the mortgage Q, the balance is reduced to 0.

So,

0 = P(1 + J)^{N}- M [ (1- (1 + J)^{N}) / J) ] M = J * [ P(1 + J)^{N}/ ((1 + J)^{N}- 1) ] M = PJ * [ (1 + J)^{N}/ ((1 + J)^{N}- 1) ] M = PJ / [ 1 - (1 + J)^{ -N}]

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Courtesy of Hans, KL Malaysia.