Amortization Schedules

One of the most important applications of annuities in business practice is repaying debts. Here, we will only consider the amortization method, which is also the most common method for repaying debts. In this method, the borrower repays the lender by a series of (usually equal) installment payments at periodic intervals. This process is called amortization of the loan. If all payments are equal and all payment intervals are equal, then payments form an annuity, and their discounted value is equal to the amount of the loan. Each payment pays the interest on the unpaid debt, and then the rest is used to repay a part of the remaining debt, thus reducing the outstanding debt and the interest paid in the following period. We start by showing how to compute the outstanding balance, that is, the outstanding debt at any given time. There are two methods on how to compute the outstanding balance: prospective method and retrospective method. Prospective method calculates the outstanding loan balance by looking into the future, that is, by finding the discounted value at that date of all the remaining installment payments. It can be applied if all the remaining payments, including the last concluding payment, are of equal size. However, that is very often not the case due to rounding, meaning that the last concluding installment payment is somewhat smaller than the previous ones, so the method cannot be applied. The second approach is the retrospective method. It calculates the outstanding loan balance looking into the past, that is, by writing the equation of value at the required date. The outstanding loan balance is calculated as the original amount of the loan accumulated to that date less the accumulated value (to that date) of all installment payments previously made. An amortization schedule is a table which shows the division of each payment into the principal and the interest part, together with the outstanding loan balance after each payment has been made. Here we show two models for an amortization schedule—the first one for level payments (the rate of interest is fixed over the term of the loan, the annuity payment periods coincide with the interest conversion periods, and all annuity payments are level) and the second one for the case of equal installments of principal (periodic payments are not level, and the amount of principal repaid in each payment remains the same).