Chapter 12A, Problem 38P

To determine

(a)

If the house is affordable or not.

Answer to Problem 38P

The house is affordable.

Explanation of Solution

Given:

The purchase price of the house is $\$170000$.

The down payment required is $5\%$.

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

The bank is offering 30 year mortgage.

Interest on mortgage is $5.35\%$ compounded annually.

The annual after tax income is $\$85000$.

Other debts are $\$850$ 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, monthly mortgage is M, 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.

$\begin{array}{c}\text{Down payment}=5\%\times \$170000\\ =\frac{5}{100}\times \$170000\\ =\$8500\end{array}$

Calculate the loan for the house.

$\begin{array}{c}\text{loan}\left(L\right)=\$170000-\$8500\\ =\$161500\end{array}$

Calculate the insurance.

$\begin{array}{c}\text{Insurance }=1\%\times \$161500\\ =\frac{1}{100}\times \$161500\\ =\$1615\end{array}$

Calculate the monthly insurance.

$\begin{array}{c}{I}^{\prime }=\frac{\text{Insurance}}{12}\\ =\frac{\$1615}{12}\\ =\$134.58\end{array}$

Calculate the monthly interest rate.

$\begin{array}{c}i=\frac{5.35\%}{12}\\ =0.4458\%\end{array}$

Calculate the monthly mortgage.

Substitute $0.004458$ for $i$, $\left(30\times 12\text{months}=360\right)$ for $n$ and $\$161500$ for $L$ in Equation (I).

$\begin{array}{c}M=\$161500\times \frac{\left[0.004458{\left(1+0.004458\right)}^{360}\right]}{\left[{\left(1+0.004458\right)}^{360}-1\right]}\\ =\$161500\times \frac{\left[0.004458\times 4.96\right]}{\left[4.96-1\right]}\\ =\$161500\times \frac{0.0221}{3.96}\\ =\$901.7\end{array}$

Calculate the monthly income.

$\begin{array}{c}I=\frac{\$85000}{12}\\ =\$7083.33\end{array}$

Calculate the debt to income ratio.

Substitute $\$901.7$ for $M$, $\$850$ for $D$, $\$134.58$ for $I^{\prime }$ and $\$7083.33$ for $I$ in Equation (II).

$\begin{array}{c}\text{Ratio}=\frac{\$901.7+\$850+\$134.58}{\$7083.33}\\ =\frac{\$1886.28}{\$7083.33}\\ =0.266\end{array}$.

Conclusion:

The house is affordable because the calculated debt to income ratio of $0.266$ which is less than the maximum debt to income ratio of $0.43$.

To determine

(b)

The total payment paid after 30 years.

Answer to Problem 38P

The total payment after $30$ years is $\$333112$.

Explanation of Solution

Write the expression for total payment.

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

Here, the total payment is P.

Calculations:

Calculate the total payment after $30$ years.

Substitute $\left(30\times 12\text{months}=360\right)$ for $n$, $\$901.7$ for $M$ and $\$8500$ for down payment in Equation (II).

$\begin{array}{c}P=\left(\$901.7\times 360\right)+\$8500\\ =\$324612+\$8500\\ =\$333112\end{array}$.

Conclusion:

The total payment after $30$ years is $\$333112$.