Question

A loan is repaid by level payments of 1 at the end of each year for 10 years. At the time of the 4th regular payment, the borrower makes an additional payment equal to the amount of principal that, according to the original payment schedule, would have been repaid in the 5 th regular payment. Payments of 1 continue to be made in the 5th and succeeding years until the loan is fully repaid. Determine the interest saved over the term of the loan. (A) 1−v 5(B) 1−v 6(C) 1−v 7(D) i(v 6+v 7 ) (E) None of the above.

Answer 1

The interest saved on the loan, when an additional payment is made to cover the principal of the next period, is the interest on that principal for one period. This is calculated as the difference between the present value factors for the 5th and 6th periods, which is (B) 1 - v to the power of 6.

Step-by-step explanation:

The student asks about the interest saved over the term of a loan when an additional payment is made equal to the amount of principal that would have been repaid in the next period, according to the original payment schedule. Specifically, this additional payment occurs at the time of the 4th regular payment, intended to cover the principal of the 5th payment, while regular payments continue after that.

To determine the interest saved, we need to find the present value of the interest that would otherwise have been paid on the principal portion of the 5th payment, which is being repaid early. Let 'v' represent the present value factor (v = 1/(1+i), where 'i' is the interest rate), and 'vn' represents the present value factor for 'n' periods. For instance, v5 is the present value factor for 5 periods.

When the principal that would have been repaid in the 5th payment is repaid one period earlier, the interest saved is the interest on this principal for one period, which is the difference between its values at the 5th and 6th periods: 1 - v6. Hence, the interest saved is represented by the formula (B) 1 - v6.