Final answer:
To determine the duration to pay off a $69,000 debt with a starting monthly payment of $1,000, increasing by $100 month-on-month at 18% annual interest compounded monthly, an amortization schedule would be needed due to the complexity of varying payments and compound interest.
Step-by-step explanation:
The question asks how long it will take to pay off a $69,000 debt with an initial monthly payment of $1,000, which increases by $100 after each payment, at a nominal interest rate of 18%, compounded monthly. This problem involves an increasing annuity due to the escalating payments and the application of the concept of present value. Calculating the exact number of months requires a complex amortization formula that takes into account both the increasing payments and the compounding interest. As each payment is made, the amount owed decreases, but the subsequent increase in the payment accelerates the payoff. With a nominal interest rate of 18% compounded monthly, significant interest is added to the balance monthly, which can extend the duration needed to fully repay the debt. Solving this accurately would typically require financial calculator or spreadsheet software that can handle the varying payment schedule coupled with the compounding interest.
For simple loans, one can use the formula to find the number of payments required to pay off a debt:
PV = R * [1 - (1 + i)^-n]/i, where PV is the present value of the loan, R is the monthly payment, i is the monthly interest rate, and n is the number of payments. However, because the payment amount increases over time in this question, this formula does not apply directly.
A practical approach to this problem involves creating an amortization schedule that tabulates payment periods, payment amounts, interest charges, and remaining balance until the entire loan is repaid. The payment increase leads to a non-standard amortization schedule, making it challenging to use a single formula to calculate the exact time to pay off the debt.
Illustrative example calculations, such as those found in Example A or B, often simplify the payment structure so that a fixed monthly payment is assumed. Yet, for this problem, the variability in payments requires a different methodology. It is a vivid reminder that paying off debt, especially with high interest rates, can be a complex and expensive endeavor over time.