## Derivation of Mortgage Loan Payment Formula

#### 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 ]
```