# Mortgage Math 101

## Mortgage Math 101

### Principal and Interest

When you get a loan, the amount of money you borrow is known as the “principal balance” of the loan. In order for the bank to make money by lending you money, you agree to pay the bank “interest” which is a percentage of the “balance.” The loan balance is the total amount owed each month, and the interest is the amount the bank charges you each month for you to rent its money.

In this discussion, we are going to pretend that you have a home mortgage loan of \$100,000 with a constant interest rate (“fixed rate“) of 5% per year, which you agree to pay over the 30-year “term” of the loan. We will also assume that you have a normal home loan with payments which must be made each month. Some loans, for example on a farm where a farmer’s income for the whole year comes in when the crops are sold, have payments once per year, but the normal home loan has monthly payments (although a few home loans have two payments per month).

For a 30-year, \$100,000 loan with a fixed 5% interest rate and monthly payments, the bank will divide the 5% by 12 in order to calculate the amount of interest each month.

One-twelfth of 5% is 0.4166666% (where the 6s really go on forever). In other words, multiplying the \$100,000 by one-twelfth of 5% gives \$416.67. The bank will round up the last 2/3 of a penny because you probably would not care, but the banks care a great deal about every little fraction of a penny.

This means that \$416.67 of your first month’s mortgage payment is needed just to pay the interest, and the remainder of the payment is going to pay down the principal balance.

For this 30 year home loan, the total monthly mortgage payment amount would be \$536.82 per month, of which \$416.67 would be paid to interest, and the remaining \$120.15 is paid to lower the \$100,000 that was borrowed. So, at the end of the first month, the \$100,000 has increased to \$100,416.67 due to interest, and once the \$536.82 payment is subtracted, the remaining amount owed to the bank is \$99,879.85.

The second month starts with less money (\$99,879.85) owed to the bank than in the first month (\$100,000) because \$120.15 has been paid toward the loan balance. In the second month, instead of the interest being \$416.67, the interest amount is now \$416.17 (actually about 3/10 of a penny less, but the bank will be happy to take that, too). So of the second month’s payment of \$536.82, the balance is paid down by \$120.65. This is fifty cents more than in the first month and at the end of the second month the amount owed to the bank is \$99,759.20.

As time goes on, month by month, the part of the \$536.82 monthly payment which goes to pay the bank interest decreases (because the amount of money being rented from the bank decreases), and the part going to actually pay off the original \$100,000 loan increases, until in the last payment the amount going to interest is about \$2.24 and the remaining \$534.58 pays off the loan.

Late Charges. Of course, if you miss a payment, then at the end of the month there is no monthly payment to pay the interest and to lower the balance, so the balance will not decrease that month. Depending on your loan paperwork, banks can either add the unpaid interest onto your account balance so that the next month’s interest includes interest on the interest you didn’t pay, or they will not charge you interest on the unpaid interest and charge you a “late charge” if you do not make the payment before a particular day in the month. The period of time after the loan payment day before the bank charges you a late charge is called the “grace period” of the loan, and most loans have a late charge of 5% of the payment which is late (about \$26.84 for the \$536.82 monthly payments in our example). Some especially greedy banks charge interest on the unpaid interest and charge a late charge.

### Fully-Amortized Loans

In the past, most home loans were “fully amortized” over 30 or 40 years, with fixed interest rates and identical monthly payments. A fully-amortized loan is one where the entire amount borrowed, the entire principal balance, is paid off at the end of the loan with fairly similar payments, and without requiring a large one-time final payment.

The example above is a 30-year fully-amortized fixed-rate loan where the \$536.82 monthly payments will pay off the entire interest and principal in 30 years. The monthly payments are each \$536.82. There are other types of loans where the payments vary because of changes in the interest rate, but as long as there is not a large final payment at the end of the loan, the loan is generally referred to as a fully-amortized loan. Loans with a large payment which must be paid on the due date of the loan are said to be “balloon-payment” loans.

Beware of a nasty trick. Some loans are calculated as if they were fully-amortized fixed-rate loans for a normal period of time, but they have a due date earlier than the time needed to pay them off. This means the bank gets the earlier payments with higher interest amounts and requires you to refinance the loan before the payments which are mostly principal.

Going back to the example of the 30-year fully-amortized 5% interest fixed-rate loan for \$100,000 we started with. The payments are \$536.82 per month. If the bank sets up the loan so that the loan is due in 20 years instead of 30 years, the amount needed to pay off the loan at that time is \$50,612. In other words, after paying \$536.82 for twenty years, you still owe the bank over half of your original loan amount! In the first two-thirds of your loan, you have paid less than one-half of it off, and if your “30-year amortized loan” is due in 20 years, after 20 years you will have a balloon payment of the other half.

### Other Types of Amortization

Going back to real 30-year loans, sometimes your monthly mortgage payments are not the exact amount needed to pay the loan off in 30 years? What happens then?

Interest-Only Loans. If your total mortgage payment is only enough to pay the interest that comes due each month, your mortgage is known as an “interest-only” loan. Since you are only paying enough to pay the interest that the bank charges each month. At the end of the loan term, you will still owe the entire original balance of the loan. Because none of your payments went toward the principal – you simply rented the money for the whole length of the loan – you still owe just as much as when you began and have the entire amount that you borrowed as a balloon payment.

Wachovia Bank was famous for its Pick-a-Pay Loans, where the borrower could select from various options what the monthly payment would be. One of the Pick-a-Pay options that was very attractive to cash-strapped homeowners was an interest-only payment, and without realizing it, by choosing the most-affordable option provided by Wachovia, thousands of homeowners found out later that they weren’t paying off their home loans at all. They were simply renting their mortgage loan money and each month as the time came closer when their loan would come due at the end of its term, the monthly payment that they would have to pay in order to really pay off their home mortgage was rapidly growing to a point where it was impossible for them to pay it.

Using the example of the \$100,000, 5% interest, 30-year loan we started with, if we pay only the \$416.67 interest in the first month, at the end of the first month the loan balance has not changed. It is still the original \$100,000, so in the second month the interest would be another \$416.67. If we pay this same \$416.67 each month for five years, because in our Pick-a-Pay loan the contract gives us this choice, and \$416.67 is easier to afford than the \$536.82, we find that at the end of five years we only have 25 years left before our mortgage has to be paid off.

Most mortgages have a specific due date when the loan must be paid off, and the loan paperwork has a requirement that if any part of the loan is not paid before that due date, whatever remains unpaid must be paid on the due date. In the normal fixed-rate fully-amortized loan, the payments are set at an amount, \$536.82 in our \$100,000 loan example, where the last month’s payment completely pays off the loan on the due date.

In the Pick-a-Pay loan example, where we pick the \$416.67, interest-only, monthly payment, and we make that payment for the first five years, at the end of the first five years we only have 25 years left in which to pay off the entire \$100,000 balance of our loan. If we decide to pay this amount over the remaining 25 years of the loan in equal payments, our monthly payment has to jump up from \$416.67 to \$584.59, which is higher than the \$536.82 which we could have paid if we had known that we were going to be trapped like this. If we wait ten years, our \$416.67 payment has to jump up to \$659.96 – the longer we pay the interest-only payments, the higher the payments will have to be once we start to pay off the principal of the loan.

That’s why some borrowers who were tricked into Pick-a-Pay loans sued Wachovia in a class action lawsuit in 2009. They settled the lawsuit, but many people who were supposed to receive benefits did not even know about the settlement. As a result, the settlement did not help many homeowners.

Negative Amortization. If your total mortgage payment isn’t even enough to pay off the interest due for that month, then your mortgage is known as a “negatively amortized” loan. Negatively amortized monthly loan payments leave you owing the bank more money at the end of each month than you owed at the beginning of the month. This is because you now owe the original principal balance plus the unpaid interest, which comes due each month, and which is not being paid by your monthly payment.

Going back to our example above, if your contract with the bank allowed you only pay only \$250.00 the first month, the balance due would then equal \$100,166.67 because only \$250 of the \$416.67 of the interest that came due that month was paid by your payment, and the other \$166.67 of interest is now owed on top of the original \$100,000.

Then the interest that you owe for the second month will actually be more than the \$416.67 for the first month because in the second month the interest is based on the new higher loan balance of \$100,166.67 which you now owe. At 5% interest, the next month’s interest payment would be \$417.36.

If you pay another \$250 in the second month, then at the end of the second month the balance has increased to \$100,334.03. The longer you pay less than the interest which comes due each month, the higher the loan balance gets, and the worse your loan situation gets.

In order to increase the amount of money that banks expected to receive from their mortgage loans each month over time, as well as in order to make mortgage loan payments more attractive to prospective borrowers, many mortgage lenders set up home loans with negative amortization for the early period of the loans. These loans frequently have low payments for the first two-to-ten years, and after that time, the monthly loan payments would be recalculated in order to pay both the accruing interest and part of the principal.

Many people did not know to look to see whether their mortgage payments would increase over time, and the loan “experts” who earned large commissions by selling them these terrible loans somehow skipped over explaining what would happen in the future. What was important to the borrowers was that they could buy the home they wanted and the payments were affordable. The fact that their mortgage was a time-bomb just ticking away was hidden from them at the time.

The banks’ explanation was that when these loans converted from negative amortization to full amortization, the borrowers would be able to pay the higher payments somehow (magic?), and the banks also figured that all of the other borrowers who would be unable to make their higher payments would come back to their friendly banks to refinance their homes. Either way, the banks expected to make a lot of money.

Of course, the banks were not counting on the borrowers being unable to pay the increased payments and the value of their homes dropping to the point where no bank would refinance these loans because the amount owed on the property was much greater than its value.

### Variable-Interest Rate Loans

Interest-only and negatively amortizing payments can be confusing, but it can be even more complicated if your interest rate and loan payments change over time.

If your mortgage does not have a fixed interest rate and identical payments for the duration of the loan, then your mortgage note will list the specific dates and change schedules when your interest rate and monthly payment amounts will change. These two types of changes may not occur at the same time, which can temporarily change the amortization of your mortgage.

For example, let’s say you have a \$500,000 loan with a 5% interest rate and a term of 30 years.  For the first month, \$2,083.33 of interest accrues. If your note sets payments of \$2,083.33 for the full year, but during that year there is an “Interest Rate Change Date” that increases the interest rate to 6%, then your loan will be negatively amortizing until your monthly payment increases to be at least equal to the interest accruing each month. Under the terms of your note, you will continue to make payments of \$2,083.33, but each month the interest accruing on the loan will be \$2,500.00 and your mortgage will increase by \$416.67.

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