Chapter 4, Problem 66QP

Summary Introduction

To calculate: The annual percentage rate and the effective annual rate of the two loans

Introduction:

Explanation of Solution

Time line of the payments:

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period. The payment is found using the formula of the present value of annuity.

Compute the present value annuity:

  $\begin{array}{c}\text{Present value annuity}=\text{C}\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$227,400=\text{C}\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\left(\frac{0.045}{12}\right)\right)}^{360}}\right)\right]}{\frac{0.045}{12}}\right\}\\ \$227,400=\text{C}\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.00375\right)}^{360}}\right)\right]}{0.00375}\right\}\\ \$227,400=\text{C}\left\{\frac{\left[1-0.259895654\right]}{0.00375}\right\}\end{array}$

    $C=\$1,276.92$

Hence, the payment C is $1,276.92

Note: Using the calculated C amount in the equation of the present value of annuity and with the original amount that has to be borrowed $225,000, the r values is found.

Time line of the payments:

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity:

  $\begin{array}{c}\text{Present value annuity}=\text{C}\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$225,000=\$1,276.92\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^{360}}\right)\right]}{r}\right\}\end{array}$

Note: To find the interest rate, it is necessary to solve the equation using a spreadsheet.

Compute the interest rate using the spreadsheet:

Step 1:

• Type the formula of the present value annuity in H6 in the spreadsheet and consider the r value as H7

Step 2:

• Assume the r value as 0.10%

Step 3:

• In the spreadsheet, go to Data and select What-If-Analysis.
• Under What-If-Analysis tab, select Goal Seek
• In set cell, select H6 (the formula)
• The ‘To value’ is considered as 225,000 (the value of the present value of annuity)
• The H7 cell is selected for the 'by changing cell'.

Step 4:

• Following the previous step, click OK in the Goal Seek Status. The Goal Seek Status appears with the r value

Step 5:

• The r value appears to be 0.4579%

Hence, the r value is 0.4580%

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=12\left(0.4580\%\right)\\ =5.50\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate with the number of periods in a year.

Hence, the annual percentage rate is 5.50%

Formula to calculate the effective annual rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.004580\right)}^{12}-1\\ =0.0564\%\end{array}$

Hence, the effective annual rate is 0.0564% or 5.64%

Calculations for the nonrefundable fee:

Annual percentage rate is 5.50%

Note: The annual percentage rate for the nonrefundable fee is the quoted rate, as the fee is not considered as a part of the loan. The effective annual rate is computed with the help of the annual percentage rate.

Formula to calculate the effective annual rate:

$\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)$

Compute the effective annual rate:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.045\right)}^{12}-1\\ ={\left(1.045\right)}^{12}-1\\ =0.0554\%\end{array}$

Hence, the effective annual rate for the non-refundable fee is 0.0554% or 5.54%.