## Partial Amortization or Bite the Bullet

A common arrangement in real estate lending might call for a 5-year loan with, say, a 15-year amortization. What this means is that the borrower makes a payment every month of a fixed amount based on a 15-year amortization. However, after 60 months, the borrower makes a single, much larger payment called a "balloon" or "bullet" to pay off the loan. Because the monthly payments don't fully pay off the loan, the loan is said to be partially amortized.

Suppose we have a \$100,000 commercial mortgage with a 12 percent APR and a 20-year (240-month) amortization. Further suppose the mortgage has a five-year balloon. What will the monthly payment be? How big will the balloon payment be?

The monthly payment can be calculated based on an ordinary annuity with a present value of \$100,000. There are 240 payments, and the interest rate is 1 percent per month. The payment is:

\$100,000 = C X [1 - (1/1.01240)/.01] = C X 90.8194 C = \$1,101.09

Now, there is an easy way and a hard way to determine the balloon payment. The hard way is to actually amortize the loan for 60 months to see what the balance is at that time. The easy way is to recognize that after 60 months, we have a 240 - 60 = 180-month loan. The payment is still \$1,101.09 per month, and the interest rate is still 1 percent per month. The loan balance is thus the present value of the remaining payments:

Loan balance = \$1,101.09 X [1 - (1/1.01180)/.01] = \$1,101.09 X 83.3217 = \$91,744.69

The balloon payment is a substantial \$91,744. Why is it so large? To get an idea, consider the first payment on the mortgage. The interest in the first month is \$100,000 X .01 = \$1,000. Your payment is \$1,101.09, so the loan balance declines by only \$101.09. Because the loan balance declines so slowly, the cumulative "pay down" over five years is not great.

We will close out this chapter with an example that may be of particular relevance. Federal Stafford loans are an important source of financing for many college students, helping to cover the cost of tuition, books, new cars, condominiums, and many other things. Sometimes students do not seem to fully realize that Stafford loans have a serious drawback: they must be repaid in monthly installments, usually beginning six months after the student leaves school.

Some Stafford loans are subsidized, meaning that the interest does not begin to accrue until repayment begins (this is a good thing). If you are a dependent undergraduate student under this particular option, the total debt you can run up is, at most, \$23,000. For Stafford loans disbursed after July 1, 1994, the maximum interest rate is 8.25 percent, or 8.25/12 = 0.6875 percent per month. Under the "standard repayment plan," the loans are amortized over 10 years (subject to a minimum payment of \$50).

Suppose you max out borrowing under this program and also get stuck paying the maximum interest rate. Beginning six months after you graduate (or otherwise depart the ivory tower), what will your monthly payment be? How much will you owe after making payments for four years?

Given our earlier discussions, see if you don't agree that your monthly payment assuming a \$23,000 total loan is \$282.10 per month. Also, as explained in Example 6.13,

Ross et al.: Fundamentals I III. Valuation of Future I 6. Discounted Cash Flow I I © The McGraw-Hill of Corporate Finance, Sixth Cash Flows Valuation Companies, 2002

Edition, Alternate Edition

186 PART THREE Valuation of Future Cash Flows after making payments for four years, you still owe the present value of the remaining payments. There are 120 payments in all. After you make 48 of them (the first four years), you have 72 to go. By now, it should be easy for you to verify that the present value of \$282.10 per month for 72 months at 0.6875 percent per month is just under \$16,000, so you still have a long way to go.

Of course, it is possible to rack up much larger debts. According to a 2001 article in Medical Economics, two married MDs, fresh out of med school, had a combined education debt of \$544,000! Ouch! Is there a finance doctor in the house? The smaller of the two loans had a balance of \$234,000, and the payments on just this portion were \$1,750 per month. The interest rate was 7 percent. The article says it will take 22 years just to pay off the loan. Is that right?

In this case, we have an ordinary annuity of \$1,750 per month for some unknown number of months. The interest rate is 7/12 = .5833 percent per month, and the present value is \$234,000. See if you agree that it will take about 260 months, or just under 22 years, to pay off the loan. Maybe MD really stands for "mucho debt!"