Chapter 6, Problem 75QP

a)

Summary Introduction

To calculate: The annual percentage rate and the effective annual rate.

Introduction:

The annual rate that is earned from the investment or charged for a borrowing is an annual percentage rate and it is represented as APR. Thus, the APR is calculated by multiplying the rate of interest for a year with the number of months in a year. The effective annual rate is the rate of interest that is expressed as if it were compounded once in a year.

Answer to Problem 75QP

The annual percentage rate is 416% and the effective annual rate is 5,370.60%.

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake consumers up. The store offers a week loan at the rate of interest of 8% per week. Then, after a few days, the store makes a one-week loan again at a discount interest rate of 8% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 8% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $36.05. As this is a discount interest rate, the net proceeding of Person X will be $63.95. Thus, Person X has to pay $100 for a month and the store lets Person X to pay $25 in installments fora week.

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(8\%\right)\\ =416\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate withthe number of months in a year. Here, the interest is calculated per week;therefore,the number of weeks in a year (52 weeks) is taken as the period.

Hence, the annual percentage rate is 416%.

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.08\right)}^{52}-1\\ =54.70604084-1\\ =53.70604084\end{array}$

Hence, the effective annual rate is 0.53,70604084 or 5,370.60%.

b)

Summary Introduction

To calculate: The annual percentage rate and the effective annual rate.

Introduction:

The annual rate that is earned from the investment or charged for a borrowing is an annual percentage rate and it is represented as APR. Thus, the APR is calculated by multiplying the rate of interest for a year with the number of months in a year. The effective annual rate is the rate of interest that is expressed as if it were compounded once in a year.

Answer to Problem 75QP

The annual percentage rate is 451.88% and the effective annual rate is 7,518.31%.

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake consumers up. The store offers a week loan at the rate of interest of 8% per week. Then, after a few days, the store makes a one-week loan again at a discount interest rate of 8% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 8% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $36.05. As this is a discount interest rate, the net proceeding of Person X will be $63.95. Thus, Person X has to pay $100 for a month and the store lets Person X to pay $25 in installments fora week.

Explanation:

In the discount loan, the amount that Person X gets is reduced by the discount and Person X has to pay the full principal value back. Person X receives $9.2 for each $10 as the principal value, with the discount of 8%. The weekly interest rates are calculated as follows:

$\begin{array}{c}\$10=\$9.2\left(1+r\right)\\ r=\left(\frac{\$10}{\$9.2}\right)-1\\ r=1.086956522-1\\ r=0.086956522\end{array}$

Note: The dollar values that are used above are not relevant. In other words,it can also be written as $0.92 and $1, or $92 and $100, or in any other combination that provides a similar rate of interest.

Hence, the r value is 0.0869 or 8.69%.

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(8.69\%\right)\\ =451.88\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate withthe number of months in a year. Here, the interest is calculated per week, and so the number of weeks in a year (52 weeks) is taken as the period.

Hence, the annual percentage rate is 451.88%.

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.0869\right)}^{52}-1\\ =76.18309552-1\\ =75.18309552\end{array}$

Hence, the effective annual rate is 75.18309552 or 7,518.31%

c)

Summary Introduction

To calculate: The annual percentage rate and the effective annual rate.

Introduction:

The annual rate that is earned from the investment or charged for a borrowing is an annual percentage rate and it is represented as APR. Thus, the APR is calculated by multiplying the rate of interest for a year with the number of months in a year. The effective annual rate is the rate of interest that is expressed as if it were compounded once in a year.

Answer to Problem 75QP

The annual percentage rate is 1,072.76%and the effective annual rate is 17,204.88622%.

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake consumers up. The store offers a week loan at the rate of interest of 8% per week. Then, after a few days, the store makes a one-week loan again at a discount interest rate of 8% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 8% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $36.05. As this is a discount interest rate, the net proceeding of Person X will be $63.95. Thus, Person X has to pay $100 for a month and the store lets Person X to pay $25 in installments fora week.

Explanation:

In this part, the present value of anannuity and the cash flow is given; with the help of the present value of annuity equation, the r value is found using a spreadsheet.

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. Using the formula of the present value of annuity, the interest rate is computed throughthe spreadsheet method.

Compute the present value annuity:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$63.95=\$25\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^4}\right)\right]}{r}\right\}\end{array}$

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 Dataand select What-If-Analysis.
• UnderWhat-If-Analysis,select Goal Seek.
• In set cell, select H6 (the formula).
• The To value is considered as 63.95 (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 20.6326657269649%.

Hence, the r value is 20.63%.

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(20.63\%\right)\\ =1,072.76\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate withthe number of periods in a year. Here, the interest is calculated per week, and so the number of weeks in a year (52 weeks) is taken as the period.

Hence, the annual percentage rate is 1,072.76%.

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.2063\right)}^{52}-1\\ =17,205.88622-1\\ =17,204.88622\end{array}$

Hence, the effective annual rate is 17,204.88622%.