Chapter 6, Problem 77QP

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 also 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 77QP

The annual percentage rate is 390%, the effective annual rate is 4,197.74%

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake-up consumers. The store offers a week loan at the rate of interest of 7.5% per week. Then, after few days, the store again makes a one-week loan at a discount interest rate of 7.5% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 7.5% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $33.55. As this is a discount interest rate, the net proceeding of Person X will be $66.45. Thus, Person X has to pay $100 for a month and the store also lets Person X to pay $25 in installments for a week.

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(7.5\%\right)\\ =390\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate with the 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 390%

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.075\right)}^{52}-1\\ =42.9774-1\\ =41.9774\end{array}$

Hence, the effective annual rate is 0.41,9774 or 4,197.74%

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 also 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 77QP

The annual percentage rate is 421.62%, the effective annual rate is 5,662.75%

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake-up consumers. The store offers a week loan at the rate of interest of 7.5% per week. Then, after few days, the store again makes a one-week loan at a discount interest rate of 7.5% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 7.5% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $33.55. As this is a discount interest rate, the net proceeding of Person X will be $66.45. Thus, Person X has to pay $100 for a month and the store also lets Person X to pay $25 in installments for a week.

Explanation:

In the discount loan, the amount that Person X gets is reduced by the discount and Person X has to pay back the full principal value. With the discount of 7.5%, Person X receives $9.25 for each $10 as the principal value. The weekly interest rates are calculated as follows:

$\begin{array}{c}\$10=\$9.25\left(1+r\right)\\ r=\left(\frac{\$10}{\$9.25}\right)-1\\ r=1.0811-1\\ r=0.0811\end{array}$

Note: The dollar values that are used above are not relevant. In other words, it can also be written as $0.925 and $1 or $92.5 and $100 or in any other combination that provides similar rate of interest.

Hence, the r value is 0.0811 or 8.11%

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(8.11\%\right)\\ =421.62\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate with the 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 421.62%

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.0811\right)}^{52}-1\\ =57.6275-1\\ =56.6275\end{array}$

Hence, the effective annual rate is 56.6275 or 5,662.75%

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 also 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 77QP

The annual percentage rate is 968.19%, the effective annual rate is 717,745.21%

Explanation of Solution

Given information:

A check-cashing store makes a personal loan to wake-up consumers. The store offers a week loan at the rate of interest of 7.5% per week. Then, after few days, the store again makes a one-week loan at a discount interest rate of 7.5% for a week. The store also makes an add-on interest on the loan at a discount interest rate of 7.5% for a week.

Thus, if Person X borrows $100 for 4 weeks, the interest would be $33.55. As this is a discount interest rate, the net proceeding of Person X will be $66.45. Thus, Person X has to pay $100 for a month and the store also lets Person X to pay $25 in installments for a week.

Explanation:

In this part, the present value of an annuity 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 formulae of the present value of annuity, the interest rate is computed using the spreadsheet method.

Compute the present value annuity:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$66.45=\$25\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^4}\right)\right]}{r}\right\}\end{array}$

Compute the interest rate using the spreadsheet:

Step 1:

• Type the formulae 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, select Goal Seek
• In set cell, select H6 (the formula)
• The To value is considered as 66.45 (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 18.6190266505771%

Hence, the r value is 18.62%

Compute the annual percentage rate:

$\begin{array}{c}\text{APR}=52\left(18.62\%\right)\\ =968.19\%\end{array}$

Note: The annual percentage rate is computed by multiplying the interest rate with the 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 968.19%

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:

$\begin{array}{c}\text{Effective annual rate}=\left(1+{\left(\frac{\text{APR}}{12}\right)}^{12}-1\right)\\ ={\left(1+0.1862\right)}^{52}-1\\ =7,178.4521-1\\ =7,177.4521\end{array}$

Hence, the effective annual rate is 7,177.4521 or 717,745.21%.