Chapter 12A, Problem 35P

To determine

(a)

If the house is affordable or not.

Answer to Problem 35P

The house is not affordable.

Explanation of Solution

Given:

The purchase price of the house is $\$260000$.

The down payment required is $10\%$.

The insurance is of $1\%$ of the loan amount each year.

The bank is offering 30 year mortgage.

Interest on the mortgage is $5.15\%$ compounded annually.

The annual after tax income is $\$55000$.

Other debts of $\$650$ per month.

The maximum debt to income ratio is $43\%$.

Concept used:

Write the expression for monthly mortgage.

$M=L\frac{\left[i{\left(1+i\right)}^n\right]}{\left[{\left(1+i\right)}^n-1\right]}$ ....... (I)

Here, loan is $L$, per payment interest is $i$ and number of payment is $n$.

Write the expression for debt to income ratio.

$\text{Ratio}=\frac{M+D+I^{\prime }}{I}$ ....... (II)

Here, monthly income is $I$, monthly debt is $D$ and monthly insurance is $I^{\prime }$.

Calculations:

Calculate the down payment for the house.

$\begin{array}{c}\text{Down payment}=10\%\times \$260000\\ =\frac{10}{100}\times \$260000\\ =\$26000\end{array}$

Calculate the loan for the house.

$\begin{array}{c}\text{loan}\left(L\right)=\$260000-\$26000\\ =\$234000\end{array}$

Calculate the insurance.

$\begin{array}{c}\text{Insurance }=1\%\times \$234000\\ =\frac{1}{100}\times \$234000\\ =\$2340\end{array}$

Calculate the monthly insurance.

$\begin{array}{c}{I}^{\prime }=\frac{\text{Insurance}}{12}\\ =\frac{\$2340}{12}\\ =\$195\end{array}$

Calculate the monthly interest rate.

$\begin{array}{c}i=\frac{5.15\%}{12}\\ =0.429\%\end{array}$

Calculate the monthly mortgage.

Substitute $0.00429$ for $i$, $\left(30\times 12\text{months}=360\right)$ for $n$, and $\$234000$ for $L$ in Equation (I).

$\begin{array}{c}M=\$234000\times \frac{\left[0.00429{\left(1+0.00429\right)}^{360}\right]}{\left[{\left(1+0.00429\right)}^{360}-1\right]}\\ =\$234000\times \frac{\left[0.00429\times 4.672\right]}{\left[4.672-1\right]}\\ =\$234000\times \frac{0.02}{3.672}\\ =\$1227.37\end{array}$

Calculate the monthly income.

$\begin{array}{c}I=\frac{\$55000}{12}\\ =\$4583.33\end{array}$

Calculate the debt to income ratio.

Substitute $\$1227.37$ for $M$, $\$650$ for $D$, $\$195$ for $I^{\prime }$ and $\$4583.33$ for $I$ in Equation (II)

$\begin{array}{c}\text{Ratio}=\frac{\$1227.37+\$650+\$195}{\$4583.33}\\ =\frac{\$2072.37}{\$4583.33}\\ =0.452\end{array}$.

Conclusion:

The house is not affordable as the calculated debt to income ratio of $0.452$ is more than the maximum debt to income ratio of $0.43$.

To determine

(b)

The total payment paid after 30 years.

Answer to Problem 35P

The total payment after $30$ years is $\$538053.2$.

Explanation of Solution

Concept used:

Write the expression for total payment.

$P=\left(M\times n\right)+\left(I^{\prime }\times 30\right)+\text{down payment}$ ....... (II)

Calculations:

Calculate the total payment after $30$ years.

Substitute $\left(30\times 12\text{months}=360\right)$ for $n$, and $\$1227.37$ for $M$, $\$2340$ for $I^{\prime }$ and $\$26000$ for down payment in Equation (II).

$\begin{array}{c}P=\left(\$1227.37\times 360\right)+\left(\$2340\times 30\right)+\$26000\\ =\$441853.2+\$70200+\$26000\\ =\$538053.2\end{array}$.

Conclusion:

Thus, the total payment after $30$ years is $\$538053.2$.