Problem Statement
Marcus takes out a mortgage of $\$300,000$ to buy an apartment. The loan has an annual interest rate of $4.8\%$ compounded monthly, and is to be repaid in equal monthly payments over $25$ years.
(a) Calculate Marcus's monthly repayment.
(b) Calculate the total interest Marcus will pay over the full duration of the loan.
Verified Solution & Marking Scheme
Part (a): TVM Parameters
Total payments $N = 25 \times 12 = 300$ months.
Monthly interest rate $r = \frac{4.8\%}{12} = 0.4\% = 0.004$ per month.
Present value $PV = 300,000$, Future value $FV = 0$.
Monthly repayment formula:
$$PMT = PV \frac{r(1 + r)^N}{(1 + r)^N - 1} = 300,000 \times \frac{0.004(1.004)^{300}}{(1.004)^{300} - 1}$$
$$(1.004)^{300} \approx 3.31016$$
$$PMT = 300,000 \times \frac{0.004 \times 3.31016}{3.31016 - 1} = 300,000 \times \frac{0.0132406}{2.31016} \approx \$1,719.44$$
Part (b): Total Interest Paid
$$\text{Total Repayments} = 300 \times 1,719.44 = \$515,832$$
$$\text{Total Interest} = \text{Total Repayments} - PV = 515,832 - 300,000 = \$215,832$$