Question

A sailboat costs $22,205. You pay 25% down and amortize the rest with equal monthly payments over a 13-year period. If you must pay 8.1% compounded monthly, what is your monthly payment? How much interest will you payMorthly payments: $(Round to two decimal places)

Answer 1

The amortization formula is given by

$PV=PMT\times(\frac{1-(1+\frac{r}{n})^{-nt}}{\frac{r}{n}})$

Where

$\begin{gathered} PV\rightarrow Present\text{ value} \\ \text{PMT}=\text{Monthly payment} \\ r=\text{anual rate} \\ n=\text{number of compounding} \\ t=\text{ time in years} \end{gathered}$

Since there is 25% down payment, then PV will be

$22,205-(25\%\times22,205)$$\begin{gathered} PV=22,205-(\frac{25}{100}\times22,205) \\ =22205-5551.25 \\ =16653.75 \end{gathered}$

Given the following

$\begin{gathered} r=\frac{8.1}{100}=0.081 \\ \frac{r}{n}=\frac{0.081}{12}=0.00675 \\ t=13 \\ nt=12\times13=156 \end{gathered}$

Substitute the values above in the amortization formula

$\begin{gathered} PV=PMT\times(\frac{1-(1+\frac{r}{n})^{-nt}}{\frac{r}{n}}) \\ 16653.75=\text{PMT}\times(\frac{1-(1+0.00675)^{-156}}{0.00675}) \end{gathered}$$\begin{gathered} 16653.75=\text{PMT}\times\frac{1-1.00675^{-156}}{0.00675} \\ 16653.75=\text{PMT}\times\frac{1-0.3501260533}{0.00675} \\ 16653.75=\text{PMT}\times\frac{0.6498739467}{0.00675} \\ 16653.75=\text{PMT}\times96.27762173 \end{gathered}$$\text{PMT}=\frac{16653.75}{96.27762173}=172.9763335$

Hence, the monthly payment is approximately $172.98