Question

A graduate has just taken out an amortized car loan of $30,000 today. The loan has a 3.60% APR with monthly compounding. The term of the loan is 7 years. The graduate would like to pay off the loan early and plans on paying the remaining balance after the 3rd year of payments. What will be the loan balance after the 3rd year of payments?

Answer 1

Rate = 3.60% / 12 = 0.3%

Number of periods = 7 * 12 = 84

Present value = Payments * [1 - 1 / (1 + r)^n] / r

30,000 = Payments * [1 - 1 / (1 + 0.003)^84] / 0.003

30,000 = Payments * [1 - 0.777539] / 0.003

30,000 = Payments * 74.153998

Payments = 404.5635

3rd year payments period = 3 * 12 = 36

Number of periods remaining = 84 - 36 = 48

Present value = Payments * [1 - 1 / (1 + r)^n] / r

Present value = 404.5635 * [1 - 1 / (1 + 0.003)^48] / 0.003

Present value = 404.5635 * [1 - 0.866074] / 0.003

Present value = 404.5635 * 44.641858

Present value = $18,060.47

Loan balance will be $18,060.47