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Amortizing a Loan

An important use of present value concepts is in determining the payments required for an installment-type loan. The distinguishing feature of this loan is that it is repaid in equal periodic payments that include both interest and principal. These payments can be made monthly, quarterly, semiannually, or annually. Installment payments are prevalent in mortgage loans, auto loans, consumer loans, and certain business loans.

To illustrate with the simplest case of annual payments, suppose you borrow $22,000 at 12 percent compound annual interest to be repaid over the next six years. Equal installment payments are required at the end of each year. In addition, these payments must be sufficient in amount to repay the $22,000 together with providing the lender with a 12 percent return.

To determine the annual payment, R, we set up the problem as follows:

$22,000 = R[∑ 1/(1 + 0.12)t]= R(PVIFA12%,6]

In Table IV in the Appendix at the end of the book, we find that the discount factor for a sixyear annuity with a 12 percent interest rate is 4.111. Solving for R in the problem above, we have

$22,000 = R(4.111)
R = $22,000/4.111 = $5,351

Thus annual payments of $5,351 will completely amortize (extinguish) a $22,000 loan in six years. Each payment consists partly of interest and partly of principal repayment. The amortization schedule is shown in Table 3.8. We see that annual interest is determined by multiplying the principal amount outstanding at the beginning of the year by 12 percent. The amount of principal payment is simply the total installment payment minus the interest payment. Notice that the proportion of the installment payment composed of interest declines over time, whereas the proportion composed of principal increases. At the end of six years, a total of $22,000 in principal payments will have been made and the loan will be completely amortized. The breakdown between interest and principal is important because on a business loan only interest is deductible as an expense for tax purposes.