Chapter 7, Problem 89P

To determine

(a)

The value of $\text{APR}$.

Answer to Problem 89P

The value of $\text{APR}$ is $6.28\%$.

Explanation of Solution

Given:

The amount is $\$220,000$.

Time period is $30$ years.

Loan origination fees is $3\%$.

Concept used:

Write the expression to calculate the monthly payments.

$\text{Monthly}\text{payments}=PMT\left(r,n,A\right)$ ...... (I)

Here, the rate for loan is $r$, time period is $n$ and the mortgage amount is $A$.

Calculation:

Calculate the total amount of mortgage with points.

$\begin{array}{c}A=\$220,000+\left(3\%\text{of}\$220,000\right)\\ =\$220,000+\left(\frac{3}{100}\times \$220,000\right)\\ =\$220,000+\$6,600\\ =\$226,600\end{array}$

Calculate the time period in term of months.

$\begin{array}{c}\text{Time}\text{period}=30\text{years}\times \left(\frac{12\text{ months}}{1\text{ year}}\right)\\ =360\text{months}\end{array}$

Calculate the rate of interest for one month.

$\begin{array}{c}\text{Rate}=\frac{6\%}{12}\\ =0.005\end{array}$

Calculate the monthly payments with points.

Substitute $0.005$ for $r$, $360$ for $n$ and $\$226,600$ for $A$ in Equation (I).

$\begin{array}{c}\text{Monthly}\text{payment}=PMT\left(0.005,360,\$226,600\right)\\ =\frac{\$226,600}{\left(\frac{1-\frac{1}{{\left(1+0.005\right)}^{360}}}{0.005}\right)}\\ =\frac{\$226,600}{166.79}\\ =\$1358.59\end{array}$

Mortgage amount without points is $\$220,000$

Calculate the $\text{APR}$.

Substitute $1358.59$ for monthly payment, $360$ for $n$ and $220,000$ for $A$ in Equation (I).

$\begin{array}{l}\$1358.59=PMT\left(\text{APR},360,\$220,000\right)\\ \$1358.59=\frac{\$220,000}{\left(\frac{1-\frac{1}{{\left(1+\text{APR}\right)}^{360}}}{\text{APR}}\right)}\\ \frac{1-\frac{1}{{\left(1+\text{APR}\right)}^{360}}}{\text{APR}}=\frac{\$220,000}{\$1358.59}\end{array}$

Calculate the value of $\text{APR}$.

$\text{APR}=6.28\%$.

Conclusion:

Thus, the value of $\text{APR}$ is $6.28\%$.

To determine

(b)

The value of $\text{APR}$.

Answer to Problem 89P

The value of $\text{APR}$ is $7.04\%$.

Explanation of Solution

Given:

The amount is

$\$220,000$.

Time period is $30$ years.

Loan origination fees is $3\%$.

Concept used:

Write the expression to calculate the monthly payments.

$\text{Monthly}\text{payments}=PMT\left(r,n,A\right)$ ...... (II)

Here, the rate for loan is $r$, time period is $n$ and the mortgage amount is $A$.

Calculation:

Calculate the total amount of mortgage with points.

$\begin{array}{c}A=\$220,000+\left(3\%\text{of}\$220,000\right)\\ =\$220,000+\left(\frac{3}{100}\times \$220,000\right)\\ =\$220,000+\$6,600\\ =\$226,600\end{array}$

Calculate the time period in term of months.

$\begin{array}{c}\text{Time}\text{period}=6\text{years}\times \left(\frac{12\text{ months}}{1\text{ year}}\right)\\ =72\text{months}\end{array}$

Calculate the rate of interest for one month.

$\begin{array}{c}\text{Rate}=\frac{6\%}{12}\\ =0.005\end{array}$

Calculate the monthly payments with points.

Substitute $0.005$ for $r$, $72$ for $n$ and $\$226,600$ for $A$ in Equation (II).

$\begin{array}{c}\text{Monthly}\text{payment}=PMT\left(0.005,72,\$226,600\right)\\ =\frac{\$226,600}{\left(\frac{1-\frac{1}{{\left(1+0.005\right)}^{72}}}{0.005}\right)}\\ =\frac{\$226,600}{60.33}\\ =\$3756\end{array}$

Mortgage amount without points is $\$220,000$ ./

Calculate the $\text{APR}$.

Substitute $\$3756$ for monthly payment, $72$ for $n$ and $\$220,000$ for $A$ in Equation (I).

$\begin{array}{l}\$3756=PMT\left(\text{APR},360,\$220,000\right)\\ \$3756=\frac{\$220,000}{\left(\frac{1-\frac{1}{{\left(1+\text{APR}\right)}^{72}}}{\text{APR}}\right)}\\ \frac{1-\frac{1}{{\left(1+\text{APR}\right)}^{72}}}{\text{APR}}=\frac{\$220,000}{\$3756}\end{array}$

Calculate the value of $\text{APR}$.

$\text{APR}=7.04\%$.

Conclusion:

Thus, the value of $\text{APR}$ is $7.04\%$.

To determine

(c)

The graph of effective interest rate.

Answer to Problem 89P

The graph of effective interest and time is shown below.

Explanation of Solution

Given:

Amount is $\$220,000$.

Time period is $30$ years.

Loan origination fees is $3\%$.

Calculation:

The following table shows the value of time and effective interest rate.

Time to pay off mortgage	Effective interest rate
$1\text{year}$	$11.5714\%$
$2\text{year}$	$8.9248\%$
$3\text{year}$	$7.9962\%$
$4\text{year}$	$7.5259\%$
7.2361% ath>5 year
$6\text{year}$	$7.0439\%$
$7\text{year}$	$6.9061\%$
$8\text{year}$	$6.8026\%$
$9\text{year}$	$6.7220\%$
$10\text{year}$	$6.6575\%$
$11\text{year}$	$6.6048\%$
$12\text{year}$	$6.5610\%$
$13\text{year}$	$6.5329\%$
$14\text{year}$	$6.4922\%$
$15\text{year}$	$6.4648\%$

Plot the graph of effective interest and time as shown below.

Graph-(1).

Conclusion:

The graph of effective interest and time is shown below.