Question

Find the amount of each payment necessary to amortize the following loan. A company borrows $84,700 for new equipment. The company agrees to make quarterly payments for 9 years at 10% per year. Find the amount of the quarterly payment.

Answer 1

This problem can be approached using the present value of annuity formula.

The present value of annuity is given by:

$PV=P \left(\frac{1-\left(1+\frac{r}{t}\right)^{-nt}}{\frac{r}{t}} \right)$

Where:: P is equal periodic payment, r is the annual interest rate, t is the number of payments in a year and n is the number of years for the loan to be paid.

Given that a company borrows $84,700 for new equipment and that the company agrees to make quarterly payments for 9 years at 10% per year.

Thus, P = $84,700; n = 9 years and r = 10% = 0.1.

Since the payment is to be made quarterly, thus, in one year, there will be 4 payments. i.e. t = 4.

Thus, we have:

$84,700=P \left(\frac{1-\left(1+\frac{0.1}{4}\right)^{-9\times4}}{\frac{0.1}{4}} \right) \\ \\ =P\left( \frac{1-(1+0.025)^{-36}}{0.025} \right)=P\left(\frac{1-1.025^{-36}}{0.025}\right) \\ \\ =P\left(\frac{1-0.4111}{0.025}\right)=P\left(\frac{0.5889}{0.025}\right)=23.56P \\ \\ \therefore P= \frac{84,700}{23.56} =\ 3,595.65$