Answer
The total interest to be paid in options 1st and 2nd are \[\$98,000\] and \[\$223,280\]. The amount of interest that the buyer will save with the 15-year option is \[\$125,280\].
Work Step by Step
Now, calculate the amount of mortgage amount by removing the amount of down payment from price of the home, use the equation as shown below:
\[\begin{align}
& \text{Mortgage amount}=\text{Home price}-\text{Down payment} \\
& =\$160,000-\$24,000\\&=\$\text{136,000}\end{align}\]
Now, it is required to compute the monthly payment value for the $136,000 mortgage at 8% for 15 years as mentioned in option 1st. Compute the monthly payment by substituting the values in the loan payment formula as shown below:
\[\begin{align}
& PMT=\frac{P\left( \frac{r}{n} \right)}{\left( 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right)} \\
& =\frac{\$136,000\times\left(\frac{0.080}{12}\right)}{1-{{\left(1+\frac{0.080}{12}\right)}^{-12\times15}}}\\&=\frac{\$136,000\times\left(0.00667\right)}{1-{{\left(1+0.00667\right)}^{-180}}}\\&=\$1,300\end{align}\]
All the calculated figures have been used to calculate the interest amount that will be paid in 15 years, now subtract the amount of total monthly payments with the amount of mortgage, use below equation:
\[\begin{align}
& \text{Total interest paid}=\left( \begin{align}
& \text{Total of all monthly payments}- \\
& \text{Amount of the mortgage} \\
\end{align} \right) \\
& =\left( \$1,300\times180\text{Months}\right)-\$136,000\\&=\text{}\!\!\$\!\!\text{98,000}\end{align}\]
Similarly, calculate the figures for option 2nd as follows:
\[\begin{align}
& \text{Down payment}=\text{Loan amount}\times \text{Percentage of downpayment} \\
& =\$160,000\\times\0.15\\&=\$24,000\end{align}\]
Now, calculate the amount of mortgage amount by removing the amount of down payment from price of the home, use the equation as shown below:
\[\begin{align}
& \text{Mortgage amount}=\text{Home price}-\text{Down payment} \\
& =\$160,000-\$24,000\\&=\$\text{136,000}\end{align}\]
Now, it is required to compute the monthly payment value for the $136,000 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:
\[\begin{align}
& \text{PMT}=\frac{P\left( \frac{r}{n} \right)}{\left( 1-{{\left( 1+\frac{r}{n} \right)}^{-nt}} \right)} \\
& =\frac{\text{ }\!\!\$\!\!\text{136,000}\times\left(\frac{0.080}{12}\right)}{1-{{\left(1+\frac{0.080}{12}\right)}^{-12\times30}}}\\&=\frac{\text{}\!\!\$\!\!\text{136,000}\times\left(0.00667\right)}{1-{{\left(1+0.00667\right)}^{-360}}}\\&=\$998\end{align}\]
All the calculated figures have been used to calculate the interest amount that will be paid in 30 years, now subtract the amount of total monthly payments with the amount of mortgage, use below equation:
\[\begin{align}
& \text{Total interest paid}=\left( \text{Total of all monthly payments}-\text{Amount of the mortgage} \right) \\
& =\left( \$998\times36\text{0Months}\right)-\$136,000\\&=\text{}\!\!\$\!\!\text{223,280}\end{align}\]
Now, to calculate interest savings in comparison of both the options, subtract the amount total interest of both the options with each other, use below equation:
\[\begin{align}
& \text{Interest Savings}=\text{Total Interest}{{\text{t}}_{\text{1st}}}-\text{Total Interest}{{\text{t}}_{\text{2nd}}} \\
& =\$98,000-\text{}\!\!\$\!\!\text{223,280}\\&=-\$\text{125,280}\end{align}\]
Hence, the total interest to be paid in options 1st and 2nd are \[\$98,000\] and \[\$223,280\]. The amount of interest that the buyer will save with the 15-year option is \[\$125,280\].
