Answer
The total interest to be paid in options 1st and 2nd are \[\$95,800\]and \[\$155,920\]. The amount of interest that the buyer will save with the 20-year option is \[~\$60,120\].
Work Step by Step
Now, calculate the amount of mortgage amount by removing the amount of down payment from price of the cabin, use the formula as shown below:
\[\begin{align}
& \text{Mortgage Amount}=\text{Cabin Price}-\text{Down payment} \\
& =\$100,000-\$5,000\\&=\$95\text{,000}\end{align}\]
Now it is required to compute the monthly payment value for the $95,000 mortgage at 8% for 20 years as mentioned in option 1st. Compute the monthly payment by substituting the values in the loan payment formula as shown below:
\[\begin{align}
& PMT=\frac{P\left( \frac{r}{n} \right)}{\left( 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right)} \\
& =\frac{\$95,000\times\left(\frac{0.080}{12}\right)}{1-{{\left(1+\frac{0.080}{12}\right)}^{-12\times20}}}\\&=\frac{\$95,000\times\left(0.00667\right)}{1-{{\left(1+0.00667\right)}^{-240}}}\\&=\$795\end{align}\]
All the calculated figures are used to calculate the interest amount that will be paid in 20 years, then subtract the amount of total monthly payments with the amount of mortgage, use equation as shown below:
\[\begin{align}
& \text{Total interest paid}=\left( \text{Total of all monthly payments}-\text{Amount of the mortgage} \right) \\
& =\left( \$795\times24\text{0Months}\right)-\$95,000\\&=\text{}\!\!\$\!\!\text{95,800}\end{align}\]
Similarly, calculate the figures for option 2nd as follows:
First, calculate the amount of down payment using equation:
\[\begin{align}
& \text{Down payment}=\text{Loan Amount}\times \text{Percentage of Downpayment} \\
& =\$100,000\times0.05\\&=\$5,000\end{align}\]
Now, calculate the amount of mortgage amount by removing the amount of down payment from price of the cabin, use the equation as shown below:
\[\begin{align}
& \text{Mortgage Amount}=\text{Cabin Price}-\text{Down payment} \\
& =\$100,000-\$5,000\\&=\$95\text{,000}\end{align}\]
Now, it is required to compute the monthly payment value for the $95,000 mortgage at 8% for 30 years as mentioned in option 2nd.
Compute the monthly payment by substituting the values in the loan payment formula as shown below:
\[\begin{align}
& PMT=\frac{P\left( \frac{r}{n} \right)}{\left( 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right)} \\
& =\frac{\$95,000\times\left(\frac{0.080}{12}\right)}{1-{{\left(1+\frac{0.080}{12}\right)}^{-12\times30}}}\\&=\frac{\$95,000\times\left(0.00667\right)}{1-{{\left(1+0.00667\right)}^{-360}}}\\&=\$697\end{align}\]
All the calculated figures are used to calculate the interest amount that will be paid in 30 years, subtract the amount of total monthly payments with the amount of mortgage, use the equation:
\[\begin{align}
& \text{Total interest paid}=\left( \text{Total of all monthly payments}-\text{Amount of the mortgage} \right) \\
& =\left( \$697\times36\text{0Months}\right)-\$95,000\\&=\text{}\!\!\$\!\!\text{155,920}\end{align}\]
Now, to calculate interest savings in comparison of both the options, subtract the amount of total interest of both the options with each other, use below equation:
\[\begin{align}
& \text{Interest Savings}=\text{Total Interes}{{\text{t}}_{1\text{st}}}-\text{Total Interes}{{\text{t}}_{2\text{nd}}} \\
& =\$95,800-\$155,920\\&=-\$\text{60,120}\end{align}\]
Hence, the total interest to be paid in options 1st and 2nd are \[\$95,800\] and \[\$155,920\], respectively. The amount of interest that the buyer will save with the 20-year option is\[~\$60,120\].
