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*((1-(1+(r)/(n))^(-nt))/((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\%*22,205)`
`\begin{gathered} PV=22,205-((25)/(100)*22,205) \\ =22205-5551.25 \\ =16653.75 \end{gathered}`

Given the following

`\begin{gathered} r=(8.1)/(100)=0.081 \\ (r)/(n)=(0.081)/(12)=0.00675 \\ t=13 \\ nt=12*13=156 \end{gathered}`

Substitute the values above in the amortization formula

`\begin{gathered} PV=PMT*((1-(1+(r)/(n))^(-nt))/((r)/(n))) \\ 16653.75=\text{PMT}*((1-(1+0.00675)^(-156))/(0.00675)) \end{gathered}`
`\begin{gathered} 16653.75=\text{PMT}*(1-1.00675^(-156))/(0.00675) \\ 16653.75=\text{PMT}*(1-0.3501260533)/(0.00675) \\ 16653.75=\text{PMT}*(0.6498739467)/(0.00675) \\ 16653.75=\text{PMT}*96.27762173 \end{gathered}`
`\text{PMT}=(16653.75)/(96.27762173)=172.9763335`

Hence, the monthly payment is approximately $172.98