# Mortgage P & I question

Why are mortgage payments structured the way they are and how are they calculated. I am not talking about an amortization calculator. What I want to know is why, each month, does one pay a slightly greater amount of principle and a slightly smaller amount of interest. It seems to me that the amount of principle and interest should remain exactly the same over time but this is not the case.

Because each payment is greater than the amount of interest that has accrued on the outstanding principal, each payment ends up, in addition to paying all accrued interest, also reducing the principal by a little bit. Consequently, in the next period, a little less interest will accrue (because of the now smaller principal balance). Because the payment amount remains the same, the proportion of a payment that goes to principal increases as you get later and later in the term of the loan.

For any payment period, let your remaining pricincpal be P(x) - this would be read as the amount of principal remaining after the xth payment. Therefore, for a 30 year loan, P(0) would be the starting principal and P(360) would be the principal remaining after the last payment.

Let your stated interest rate be SR. Your monthly interest rate will be calculated as SR/12 - let this be written as MI.

Let your monthly payment amount be MPA.

Let the interest you pay be stated as I(x) - so the interest you pay for your 15th monthly payment would be I(15).

Let the amount of your payment, for each payment, that goes towards principal be ATP(x).

Each month:

I(x) = MI * P(x-1)
ATP(x) = MPA - I(x)

For a 30 year loan, MPA is calculated so that the sum of ATP(0), ATP(1),…,ATP(360) = P(0)

The reason that the amount of interest you pay cannot remain the same each time is because you are paying interest on the remaining loan amount, and the amount decreases if you pay it off.

If you had an interest only loan, then your loan amount would stay the same and so would your principal and interest payments (zero, and the entire payment amount, respectively).

The reason is that people want their payment to be the same so they can know their budget in advance.

To see how this works, take a simple example. Say you have a mortgage for \$100,000 at yearly interest rate of 5%. This means that each month you owe 1/12 of 5%, or .416% in interest.

So at the end of the first month when your first payment is due, you owe the original \$100,000, plus \$416.66 interest. So let’s say you pay \$450. This pays the \$416.66 in interest, and amount left over reduces the principal by \$33.34, leaving your balance at \$99,966.66.

At the end of the next month, you owe .416% interest on that balance, which is a little less than the previous month, now it’s only \$415.86. If you make the same \$450 payment, since there is a little less interest owed this month, there is a little more left over to put against the principal, \$34.14. This reduces the balance to \$99,932.52.

So each month, because the principal is lower less interest accrues. That means as long as you have a fixed payment amount and a fixed interest rate, there is a little more of each payment left over after paying the interest, which then goes toward the principal.

The trick is to figure out exactly what the fixed payment amount has to be to pay off your balance in exactly some number of the same amount payments. This is where the formulas given above come in.

Why should it be the case that the amount of principle and interest should be the same?

A mortgage, a loan or your credit card are all the same. Each month (usually) the amount of interest you owe (dollars owed times interest rate per month) is calculated and you must pay that.

The standard mortgage scheme is to have a simple single payment of a fixed amount every month (easier to budget). If that payment is bigger than the amount of interest you would owe, the rest goes to principle.

that means, next month you owe less money, you pay less interest, then more goes to principle.

Obviously, as you pay down the principle, the amount going to principle gets bigger and bigger each month, and you pay it off faster and faster; so the amount owed per month when graphed forms a curve (downward) not a straight line. Toward the end almost all of the payment is principle. The first few months, with a long amortization (say, 30 years) almost everything in the first payment is interest.