A printing firm borrows $50,000 to purchase a new printing machine. Under the loan agreement, the firm will make 12 equal monthly repayments with the first repayment made 6 months after the loan amount is transferred to the firms bank account. If the rate of interest charged on the loan is 12% per annum compounded monthly, what amount will be each of the 12 monthly repayments?
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+118680 A printing firm borrows $50,000 to purchase a new printing machine. Under the loan agreement, the firm will make 12 equal monthly repayments with the first repayment made 6 months after the loan amount is transferred to the firms bank account. If the rate of interest charged on the loan is 12% per annum compounded monthly, what amount will be each of the 12 monthly repayments?
Inital loan = $50000, interest rate = 1% per month = 0.01
The loan accrues interest for 5 months before the repayment period starts.
$$\boxed{Future \;value\; with\; compound\;interest=P(1+i)^n}$$
Effective loan = 50000(1.01)^5 = $52550.50
Now it is a present value of an ordinary annuity problem,
$$\boxed{PV=R\times \frac{1-(1+i)^{-n}}{i}}$$
$$\begin{array}{rll}
PV&=&R\times \frac{1-(1+i)^{-n}}{i}\\\\
52550.50&=&R\times \frac{1-(1.01)^{-12}}{0.01}\\\\
52550.50\times \frac{0.01}{1-(1.01)^{-12}} &=&R \\\\
R&=&\$4669.05 \\\\
\end{array}\\\\
\mbox{Each monthly payment will be \$4669.05}$$
$${\frac{\left({\mathtt{50\,000}}{\mathtt{\,\times\,}}\left({{\mathtt{1.01}}}^{{\mathtt{6}}}\right)\right)}{{\mathtt{12}}}} = {\mathtt{4\,423.000\: \!627\: \!504\: \!166\: \!666\: \!7}}$$
.+118680 A printing firm borrows $50,000 to purchase a new printing machine. Under the loan agreement, the firm will make 12 equal monthly repayments with the first repayment made 6 months after the loan amount is transferred to the firms bank account. If the rate of interest charged on the loan is 12% per annum compounded monthly, what amount will be each of the 12 monthly repayments?
Inital loan = $50000, interest rate = 1% per month = 0.01
The loan accrues interest for 5 months before the repayment period starts.
$$\boxed{Future \;value\; with\; compound\;interest=P(1+i)^n}$$
Effective loan = 50000(1.01)^5 = $52550.50
Now it is a present value of an ordinary annuity problem,
$$\boxed{PV=R\times \frac{1-(1+i)^{-n}}{i}}$$
$$\begin{array}{rll}
PV&=&R\times \frac{1-(1+i)^{-n}}{i}\\\\
52550.50&=&R\times \frac{1-(1.01)^{-12}}{0.01}\\\\
52550.50\times \frac{0.01}{1-(1.01)^{-12}} &=&R \\\\
R&=&\$4669.05 \\\\
\end{array}\\\\
\mbox{Each monthly payment will be \$4669.05}$$
