Question

8. Alvin purchased $1,300,000 worth of videogame set and made a $150,000 down payment. He agreed to pay the balance by making equal payments at the end of each month for 20 years. What is the size of the monthly payment if the interest charged is 16% compounded quarterly?

Answer 1

We have a purchase of $1,300,000.

The downpayment is $150,000 and the rest is financed.

We can calculate the amount that is owed as:

$\begin{gathered} C=1,300,000-150,000 \\ C=1,150,000 \end{gathered}$

This amount will be paid in equal amounts, monthly for 20 years.

The interest rate is 16% compounded quarterly.

We have to start by converting the interest rate in a monthly-compounded equivalent rate.

A rate of 16% compounded quarterly (m=3) will be equivalent to a monthly rate r. We can calculate r as:

$\begin{gathered} (1+r)^{12}=(1+\frac{0.16}{3})^3 \\ (1+r)^4=1+\frac{0.16}{3} \\ (1+r)^4\approx1+0.05333 \\ 1+r\approx1.0533^{\frac{1}{4}} \\ r\approx1.013-1 \\ r\approx0.013 \end{gathered}$

We then can calculate the payments as an annuity with r = 0.013 and 20*12 = 240 payments.

We can calculate the amount he will pay each month as:

$\begin{gathered} P=\frac{r\cdot PV}{1-(1+r)^{-n}} \\ P=\frac{0.013\cdot1150000}{1-(1+0.013)^{-240}} \\ P=\frac{14950}{1-1.013^{-240}} \\ P=\frac{14950}{1-0.045} \\ P=\frac{14950}{0.955} \\ P=15654.45 \end{gathered}$

Answer: the monthly payment is approximately $15654.45.