A $12,000 loan is repaid by semi-annual payments of $1,500 each. Interest on the loan is 10% compounded semi-annually. How long will it take to pay off the loan?

Question

Quizplus · Accepted Answer

The loan is repaid semi-annually, meaning twice a year. To find the number of payments, we use the formula for the present value of an annuity: $PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)$, where PV is the present value (loan amount), PMT is the payment amount, r is the interest rate per period, and n is the total number of payments. Rearranging to solve for n, we get $n = \frac{\log(1 - \frac{PV \times r}{PMT})}{\log(1 + r)}$. Plugging in the values: $PV = 12000$, $PMT = 1500$, $r = 0.05$ (since 10% annual interest compounded semi-annually gives a 5% semi-annual rate), we can solve for n. However, without needing to solve it explicitly, knowing that $1,500 is paid semi-annually means it will take 8 payments (4 years) to reach $12,000 without interest. Considering the interest, it will take more than 4 years but less than 11 years (the next possible whole number of years with semi-annual payments). The only option fitting this description is 5.5 years, implying 11 payments.

A $12,000 Loan Is Repaid by Semi-Annual Payments of $1,500