The Math Behind Your Home Loan

 TOOLBOX Resize
By Jack Guttentag
Saturday, October 6, 2007

The one thing that almost all borrowers know about their mortgages is the amount of the initial scheduled payment. This is how much they are obliged to pay each month under the terms of the mortgage contract. They know that failure to pay that amount violates the contract, leading to late charges, delinquency reports and, ultimately, foreclosure.

While borrowers know the amount, they are often hazy about how it is calculated and what it includes. I will illustrate several possibilities using a \$100,000 loan at 6 percent.

In the simplest possible case, the scheduled payment includes only interest until the final payment, when it includes repayment of the balance. The interest payment each month is 0.06 divided by 12, or 0.005, multiplied by \$100,000, which equals \$500. The final payment, assuming the borrower paid only interest throughout, would be \$100,500.

Most mortgages written in the 1920s were of this type, usually with terms of five or 10 years. Their weakness is that they must be refinanced at term, which during the Depression of the 1930s became difficult because property values and borrower incomes had fallen. The notion took hold that it was prudent for borrowers to pay down the balance over time by making a mortgage payment larger than the interest. This additional amount is called the principal payment.

The principal payment is always a residual -- that is, the total payment less the interest. If the borrower in the example paid \$600, the \$500 of interest would be deducted, leaving \$100 as the principal payment. If the borrower paid \$700, the principal payment would be \$200.

Including principal in the scheduled payment requires a rule for determining what that payment is. The most obvious rule is to pay back equal amounts of principal every month. If our sample loan is for 30 years, we divide \$100,000 by 360 to get a principal repayment of \$277.78 a month.

Loans of this type have existed, most recently in New Zealand, but they have a serious drawback. The scheduled payment that includes a fixed amount of principal every month starts high -- too high for many borrowers -- and ends low because of the decline in interest owed on the shrinking principal. In month one, the scheduled payment is \$277.78 plus \$500, or \$777.78. In month 360, it is \$277.78 plus \$1.39, or \$279.17.

So some unknown mathematician reasoned as follows: "If payments beginning at \$777.78 and declining every month to \$279.17 will pay off this loan, there must be some amount in between, which, if made every month without change, would do the same." The reasoning is correct -- the amount (\$599.56 in my example) is called the fully amortizing payment.

The fully amortizing payment is calculated using an equation that my editor says does not belong in a family publication. If you are curious, it is on my Web site, at http://www.mtgprofessor.com/formulas.htm. But you don't need the equation; while solving the equation would take minutes, financial calculators offer the answer instantly.

If the interest rate does not change, the fully amortizing payment is constant over the life of the loan. However, the part that goes to interest declines every month, and the part going to principal rises.

After World War II, almost all U.S. mortgages carried fully amortizing payments.

However, borrowers generally like lower initial payments, and lenders seek to accommodate them if possible. The only way to reduce the initial payment is to reduce the principal payment -- principal is generally the only component that has any give in it. Lenders will not forgo interest, but they may allow borrowers to delay making principal payments. Although such loans had fallen out of favor in the 1930s, they returned in recent years with interest-only and option adjustable-rate mortgages. In recent months, they have again been under great scrutiny because of concerns that such exotic loans increase the risk of payment shock and foreclosure.

On mortgages with an interest-only option, the scheduled payment is the interest payment for the length of the interest-only period, usually five to 10 years. After that, the scheduled payment becomes the fully amortizing payment.

On option ARMs, borrowers select their scheduled payment during the first five or 10 years. The can select the fully amortizing payment over either 15 or 30 years, the interest-only payment, or a "minimum" payment that is less than the interest. Most borrowers select the last, and they sometimes find themselves in trouble when their scheduled payment increases in future years.

The scheduled payment may not be limited to interest and principal. The monthly mortgage insurance premium, if there is one, will be included. If the borrower has agreed to an escrow for property taxes and homeowners insurance, most lenders treat the monthly escrow payments as if they are also part of the scheduled payment; if the escrow payment is short, the payment is considered delinquent. A borrower can start down a slippery path to foreclosure by failing to pay required escrows.

Jack Guttentag is professor of finance emeritus at the Wharton School of the University of Pennsylvania. He can be contacted through his Web site,http://www.mtgprofessor.com.