Question

Recall that the monthly payment (PMT) is constant for a fully amortizing residential mortgage. If BAL denotes the original loan balance, N denotes the number of years, and Y is the (annual) interest rate, then PMT can be obtained by solving the equation:

BAL = PMT/(1+Y/12) + PMT/(1+Y/12)² + PMT/(1+Y/12)³ + PMT/(1+Y/12)^12N

[1] Can you give a brief explanation of this equation?

It’s a straight present value calculation. It says that when the lender loans out BAL and receives the payment stream PMT (each month) then the lender’s yield equals Y.

[2] Can you sum the geometric series to get a more compact expression?

Answer 1

Can you give a brief explanation of this equation?

The loan balance at any point in time should be nothing but sum of present value of all the future payments. A borrower, under this arrangement of loan, will have to pay the monthly payment (PMT), over N number of years, and Y is the (annual) interest rate.

Hence, monthly interest rate = r = Y / 12

and total number of payments = 12 x N = 12N

Hence, BAL = original loan balance = PV of all the future payments = PMT / (1 + Y/12) + PTM / (1 + Y/12)² + PMT / (1 + Y/12)³ + .......+ PMT / (1 + Y/12)^2N

Can you sum the geometric series to get a more compact expression?

Let's assume 1/(1 + Y/12) as z

Hence, BAL = PMT x z + PMT x z² + PMT x z³ + .....+ PMT x z^2N = PMT x z x (1 + z + z² + .....+ z^2N-1)

Sum of the geometric series = (1 + z + z² + .....+ z^2N-1) = (1 - z^2N) / (1 - z)

Hence, BAL = PMT x z x (1 - z^2N) / (1 - z)

Substitute z = 1 / (1 + Y / 12)