Derivation of Mortgage Loan Payment Formula

(Sent to me by "Hans" Gurdip Singh)


Concept

  1. Calculate H = P*J, this is your current monthly interest
  2. 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.
  3. Calculate Q = P - C, this is the new balance of your principal of your loan.
  4. 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 rn - 1

so the sum of the series is expressed as

Sn = a [ (1 - rn ) / ( 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

  Sn = 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.