㵅阳大学
UNIVERSIDADE DE MACAU
8. Mr. Chan has to pay for a loan with equal payments of $\$ 1500$ at the beginning of each month for 25 years. If the interest rate is $\overline{11.5} \%$ p.a. compounded monthly, calculate:
(a) the total amount repaid, (5\%)
(b) the present value of the loan (correct to 2 decimal places), ( $10 \%$ )
(c) the total amount of interest paid (correct to 2 decimal places). (5\%)
i) Let the amount repaid monthly be $a$.
$$
\begin{array}{l}
\text { 1. } \
500=a\left(1+\frac{11.5 \%}{12}\right)^{1}+a\left(1+\frac{11.5 \%}{12}\right)^{2}+\cdots a\left(1+\frac{11.5}{12}\right)^{30}
\end{array}
$$

Question

㵅 阳大学
UNIVERSIDADE DE MACAU
8. Mr. Chan has to pay for a loan with equal payments of $\$ 1500$ at the beginning of each month for 25 years. If the interest rate is $\overline{11.5} \%$ p.a. compounded monthly, calculate:
(a) the total amount repaid, (5\%)
(b) the present value of the loan (correct to 2 decimal places), ( $10 \%$ )
(c) the total amount of interest paid (correct to 2 decimal places). (5\%)
i) Let the amount repaid monthly be $a$.
$$
\begin{array}{l}
\text { 1. } \
500=a\left(1+\frac{11.5 \%}{12}\right)^{1}+a\left(1+\frac{11.5 \%}{12}\right)^{2}+\cdots a\left(1+\frac{11.5}{12}\right)^{30}
\end{array}
$$

Accepted Answer

To solve Mr. Chan's loan problem, we can break it down into the parts outlined in the question. ### Given Information: - Monthly payment: $1,500 - Interest rate: 11.5% per annum compounded monthly - Loan term: 25 years (which is 25 * 12 = 300 months) ### (a) Total Amount Repaid The total amount repaid over the loan term can be calculated as: $$ \text{Total Amount Repaid} = \text{Monthly Payment} \times \text{Total Number of Payments} $$ $$ \text{Total Amount Repaid} = 1,500 \times 300 = 450,000 $$ ### (b) Present Value of the Loan To find the present value (PV) of the loan, we can use the formula for the present value of an annuity: $$ PV = P \times \left(1 - (1 + r)^{-n}\right) / r $$ Where: - $ P $ = monthly payment = $1,500 - $ r $ = monthly interest rate = $ \frac{11.5\%}{12} = \frac{0.115}{12} \approx 0.00958333 $ - $ n $ = total number of payments = 300 Substituting the values: $$ PV = 1,500 \times \left(1 - (1 + 0.00958333)^{-300}\right) / 0.00958333 $$ Calculating this will give you the present value of the loan. ### (c) Total Amount of Interest Paid The total amount of interest paid can be calculated as: $$ \text{Total Interest Paid} = \text{Total Amount Repaid} - \text{Present Value of the Loan} $$ Once you calculate the present value, you can easily find the total interest paid. If you need help with the calculations or any specific part, feel free to ask!

Sia