Chapter 17, Problem 11QP

a)

Summary Introduction

To determine: The effective annual interest rate (EAR).

Introduction:

EAR is the actual rate that is earned by an individual. These interest rates are generally shown as they are compounded once in a year.

Answer to Problem 11QP

The EAR is 20.13%.

Explanation of Solution

Given information:

A company has offered a sales term of 1/10, net 30.

Note: It means that the customers receive 1% discount if they make the payment in 10 days, with the total amount due in 30 days, if the discount is not received.

Steps to determine the EAR:

• Firstly, determine the interest rate for the term of the discount.
• Secondly, compute the interest is for how many days.
• Finally, determine the EAR using the formula.

The formula to calculate the periodic interest rate:

$\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}$

The formula to calculate the number of days of interest rate:

$\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}$

The formula to calculate the EAR:

$\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]$

Where,

m refers to the number of days.

Compute the periodic interest rate:

$\begin{array}{c}\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}\\ =\frac{\left(\frac{1}{100}\right)}{\left(1-\frac{1}{100}\right)}\\ =\frac{0.010}{\left(1-0.010\right)}\end{array}$

$\begin{array}{c}=\frac{0.010}{0.99}\\ =0.0101\end{array}$

Hence, the periodic interest rate is 0.0101 or 1.01%.

Compute the number of days of interest rate:

$\begin{array}{c}\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}\\ =30-10\\ =20\text{ days}\end{array}$

Hence, the number of days of interest rate period is 20 days.

Compute the EAR:

$\begin{array}{c}\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]\\ =\left[{\left(1+\frac{1.01}{100}\right)}^{\left(\frac{365}{20}\right)}-1\right]\\ =\left[{\left(1+0.0101\right)}^{18.25}-1\right]\\ =\left[{\left(1.0101\right)}^{18.25}-1\right]\end{array}$

$\begin{array}{c}=\left[1.201295285-1\right]\\ =0.2013\end{array}$

Hence, the EAR is 0.2013 or 20.13%.

Summary Introduction

To determine: The effective annual interest rate (EAR) if the discount rate is 2%.

Answer to Problem 11QP

The EAR is 44.56%.

Explanation of Solution

Given information:

The terms “2/10 net 30” means the customers receive 2% discount if they make the payment in 10 days, with the total amount due in 30 days (if the discount is not received).

The formula to calculate the periodic interest rate:

$\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}$

The formula to calculate the number of days of interest rate:

$\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}$

The formula to calculate the EAR:

$\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]$

Where,

m refers to the number of days

Compute the periodic interest rate:

$\begin{array}{c}\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}\\ =\frac{\left(\frac{2}{100}\right)}{\left(1-\frac{2}{100}\right)}\\ =\frac{0.020}{\left(1-0.020\right)}\end{array}$

$\begin{array}{c}=\frac{0.020}{0.98}\\ =\text{0}\text{.0204}\end{array}$

Hence, the periodic interest rate is 0.0204 or 2.04%.

Compute the number of days of interest rate:

$\begin{array}{c}\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}\\ =30-10\\ =20\text{ days}\end{array}$

Hence, the number of days of interest rate period is 20 days.

Compute the EAR:

$\begin{array}{c}\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]\\ =\left[{\left(1+\frac{\text{0}\text{.0204}}{100}\right)}^{\left(\frac{365}{20}\right)}-1\right]\\ =\left[{\left(1+0.0204\right)}^{18.25}-1\right]\\ =\left[{\left(1.0204\right)}^{18.25}-1\right]\end{array}$

$\begin{array}{c}=\left[1.445641-1\right]\\ =0.4456\end{array}$

Hence, the EAR is 0.4456 or 44.56%.

b)

Summary Introduction

To determine: The EAR when the credit period has increased to 45 days.

Answer to Problem 11QP

The EAR is 11.049%.

Explanation of Solution

Given information:

A company has offered a sales term of 1/10, net 45.

Note: It means that the customers receive 1% discount if they make the payment in 10 days, with the total amount due in 45 days if the discount is not received.

The formula to calculate the periodic interest rate:

$\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}$

The formula to calculate the number of days of interest rate:

$\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}$

The formula to calculate the EAR:

$\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]$

Where,

m refers to the number of days.

Compute the periodic interest rate:

$\begin{array}{c}\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}\\ =\frac{\left(\frac{1}{100}\right)}{\left(1-\frac{1}{100}\right)}\\ =\frac{0.010}{\left(1-0.010\right)}\end{array}$

$\begin{array}{c}=\frac{0.010}{0.99}\\ =0.0101\end{array}$

Hence, the periodic interest rate is 0.0101 or 1.01%.

Compute the number of days of interest rate:

$\begin{array}{c}\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}\\ =45-10\\ =35\text{ days}\end{array}$

Hence, the number of days of interest rate period is 35 days.

Compute the EAR:

$\begin{array}{c}\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]\\ =\left[{\left(1+\frac{1.01}{100}\right)}^{\left(\frac{365}{35}\right)}-1\right]\\ =\left[{\left(1+0.0101\right)}^{10.429}-1\right]\\ =\left[{\left(1.0101\right)}^{10.429}-1\right]\end{array}$

$\begin{array}{c}=\left[1.11049-1\right]\\ =0.11049\end{array}$

Hence, the EAR is 0.11049 or 11.049%.

c)

Summary Introduction

To determine: The EAR when the discount period has decreased to 20 days.

Answer to Problem 11QP

The EAR is 44.31%.

Explanation of Solution

Given information:

A company has offered a sales term of 1/10, net 30.

The formula to calculate the periodic interest rate:

$\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}$

The formula to calculate the number of days of interest rate:

$\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}$

The formula to calculate the EAR:

$\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]$

Where,

m refers to the number of days.

Compute the periodic interest rate:

$\begin{array}{c}\text{Periodic interest rate}=\frac{\text{Discount}\text{rate offered}}{\left(1-\text{Discount}\text{rate}\right)}\\ =\frac{\left(\frac{1}{100}\right)}{\left(1-\frac{1}{100}\right)}\\ =\frac{0.010}{\left(1-0.010\right)}\end{array}$

$\begin{array}{c}=\frac{0.010}{0.99}\\ =0.0101\end{array}$

Hence, the periodic interest rate is 0.0101 or 1.01%.

Compute the number of days of interest rate:

$\begin{array}{c}\text{Number of days of interest rate}=\text{Credit period}-\text{Period of discount}\\ =30-20\\ =10\text{ days}\end{array}$

Hence, the number of days of interest rate period is 10 days.

Compute the EAR:

$\begin{array}{c}\text{EAR}=\left[{\left(1+\text{Periodic rate}\right)}^m-1\right]\\ =\left[{\left(1+\frac{1.01}{100}\right)}^{\left(\frac{365}{10}\right)}-1\right]\\ =\left[{\left(1+0.0101\right)}^{36.5}-1\right]\\ =\left[{\left(1.0101\right)}^{36.5}-1\right]\end{array}$

$\begin{array}{c}=\left[1.4431-1\right]\\ =0.4431\end{array}$

Hence, the EAR is 0.4431 or 44.31%.