1. Suppose that you have a $15,000 balance on a car loan. The balance accrues interest annually at a rate of 7% of the total unpaid balance at the end of the year. So the balance in one year depends on the current balance, the interest rate, and the payment:
New balance = (Previous balance − payment) · (1 + i) (1)
For example, if you make a $2,500 payment in the first year, then your balance next year will be ($15,000 − $2,500) · 1.07 = $13,375. Note that if you make the same payment each year, then after T years, the balance will be:1
Year T balance = (Initial balance) · (1 + i)T − payment · (1 + i)T+1 − (1 + i) (3) i
Answer the following:
(a) Ignoring any minimum payments required by the loan contract, what is the lowest payment that you could make in the initial year such that the balance in the next year is not greater than $15,000?
(b) Suppose that you commit yourself to paying $2,000 per year (or the remaining balance; which ever is less) until the loan is paid off.
How many years will it take you to pay of the loan?
What is the balance of the loan in the year before it is paid off?
What is the cumulative dollar value of the interest that you will end up paying to your lender?
(c) Suppose that you wish to pay off your loan in 5 years.
What is the minimum annual payment required?
What is the cumulative dollar value of the interest that you will end up paying to your lender?