Chapter 7, Problem 88P

To determine

(a)

The annual percentage rate.

Answer to Problem 88P

The annual percentage rate is $9.4\%$.

Explanation of Solution

Given:

The loan amount is $\$18000$.

The time period is $60\text{ months}$.

The interest rate is $9\%$.

Concept used:

Write the expression for loan origination fee of $2$ points.

$F=2\%\times A$ ......(I)

Here, the loan origination fee is F and the loan amount is $A$.

Write the expression for the interest paid.

$I=i\times A$ ......(II)

Here, the interest paid is I and the interest rate is $i$.

Write the expression for the total cost of the loan.

$L=F+I$ ......(III)

Here, the total cost of the loan is F.

Write the expression for the Annual Percentage Rate.

$APR=i\times \frac{L}{I}$ ......(IV)

Here, the annual percentage rate is APR.

Calculations:

Calculate the loan origination fee of $2$ points.

Substitute $\$18000$ for $A$ in Equation (I).

$\begin{array}{c}F=2\%\times \$18000\\ =\frac{2}{100}\times \$18000\\ =\$360\end{array}$

Calculate the interest paid in $5$ years.

Substitute $\$18000$ for $A$ and $9\%$ for $i$ in Equation (II).

$\begin{array}{c}I=9\%\times \$18000\\ =\frac{9}{100}\times \$18000\\ =\$8100\end{array}$

Calculate the total cost of the loan.

Substitute $\$8100$ for $I$ and $\$360$ for $F$ in Equation (III).

$\begin{array}{c}L=\$8100+\$360\\ =\$8460\end{array}$

Calculate the Annual Percentage Rate.

Substitute $\$8100$ for $I$, $9\%$ for $i$ and $\$8460$ for $L$ in Equation (IV).

$\begin{array}{c}APR=9\%\times \frac{\$8460}{\$8100}\\ =\frac{9}{100}\times \frac{\$8460}{\$8100}\\ =9.4\%\end{array}$.

Conclusion:

Thus, the value of annual percentage rate is $9.4\%$.

To determine

(b)

The annual percentage rate.

Answer to Problem 88P

The annual percentage rate is $23.5\%$.

Explanation of Solution

Given:

The loan amount is $\$18000$.

The time period is $24\text{months}$.

The interest rate is $9\%$.

Concept used:

Write the expression for the interest rate.

$i=\frac{I^{\prime }}{A}\times 100$ ......(V)

Here, the interest for each year is $I^{\prime }$.

Calculations:

Calculate the loan origination fee of $2$ points.

Substitute $\$18000$ for $A$ in Equation (I).

$\begin{array}{c}F=2\%\times \$18000\\ =\frac{2}{100}\times \$18000\\ =\$360\end{array}$

Calculate the interest paid in $2$ years.

Substitute $\$18000$ for $A$ and $9\%$ for $i$ in Equation (II).

$\begin{array}{c}I=9\%\times \$18000\\ =\frac{9}{100}\times \$18000\\ =\$8100\end{array}$

Calculate the total cost of the loan.

Substitute $\$8100$ for $I$ and $\$360$ for $F$ in Equation (III).

$\begin{array}{c}L=\$8100+\$360\\ =\$8460\end{array}$

Calculate the interest for each year.

$\begin{array}{c}{I}^{\prime }=\frac{\$8460}{2}\\ =\$4230\end{array}$

The interest for each year is $\$4230$.

Calculate the annual percentage rate.

Calculate the interest percentage for each year.

Substitute $\$4230$ for $I^{\prime }$ and $\$18000$ for $A$ in Equation (V).

$\begin{array}{c}i=\frac{\$4230}{18000}\times 100\\ =23.5\%\end{array}$.

Conclusion:

Thus, the value of annual percentage rate is $23.5\%$.

To determine

(c)

The graph for the effective interest rate with time to sell the car and pay-off the loan.

Explanation of Solution

The graph of effective interest rate with time to sell car and pay-off the loan is shown below.