Chapter 19, Problem 8QP

a)

Summary Introduction

To compute: The decrease in the outstanding balances of cash due to the implementation of lockboxes.

Introduction:

A lockbox is a distinct post office box that the companies use to speed up the collection of accounts receivable. The purpose of the lockbox is to reduce the collection time of accounts receivable.

Answer to Problem 8QP

The reduction in the cash balance is $390,000.

Explanation of Solution

Given information:

The time taken to receive and deposit the checks from the customer by Company C is 6 days. In order to reduce its collection time, the management is planning for a lockbox system, which will reduce the collection time to 3 days. The average daily collections are $130,000, and the required rate of return is 5% per year.

Explanation:

Formula to compute the reduction in the cash balance:

$\text{Reduction in the cash balance}=\left[\begin{array}{l}\text{Expected reduction in the}\\ \text{ collection due to lockbox system}\left(\text{Average daily collections}\right)\end{array}\right]$

Compute the reduction in the cash balance:

$\begin{array}{c}\text{Reduction in the cash balance}=\left[\begin{array}{l}\text{Expected reduction in the}\\ \text{ collection due to lockbox system}\left(\text{Average daily collections}\right)\end{array}\right]\\ =3\left(\$130,000\right)\\ =\$390,000\end{array}$

Hence, the reduction in the cash balance is $390,000.

b)

Summary Introduction

To compute: The return on dollar on the savings.

Answer to Problem 8QP

The daily return on dollar on the savings is $52.26.

Explanation of Solution

Formula to compute the average daily rate:

$\text{Average daily rate}={\left(1+\text{Required rate of return}\right)}^{\frac{1}{365}}-1$

Compute the average daily rate:

$\begin{array}{c}\text{Average daily rate}={\left(1+\text{Required rate of return}\right)}^{\frac{1}{365}}-1\\ ={1.05}^{\frac{1}{365}}-1\\ =0.000134\%\text{ per day}\end{array}$

Hence, the average daily rate is 0.000134% per day.

Formula to compute the daily return on dollar:

$\text{Daily return on dollar}=\text{Balance in cash reduction}\left(\text{Average daily rate}\right)$

Compute the daily return on dollar:

$\begin{array}{c}\text{Daily return on dollar}=\text{Balance in cash reduction}\left(\text{Average daily rate}\right)\\ =\$390,000\left(0.000134\right)\\ =\$52.26\end{array}$

Hence, the daily return on dollar is $52.26.

c)

Summary Introduction

To compute: The maximum charge that Company C must pay for the lockbox system monthly and the maximum charge if it is due at the starting of the month.

Answer to Problem 8QP

The monthly charge if the payment happens at the end of the month is $1,588.86, and the monthly charge if the payment happens at the beginning of the month is $1,582.41.

Explanation of Solution

If the company adopts the lockbox, it will obtain three payments early. Hence, the savings are the reduction in the cash balance of $390,000.

Formula to compute the monthly rate of interest:

$\text{Monthly rate of interest}={\left(1+\text{Rate of return}\right)}^{\frac{1}{12}}-1$

Compute the monthly rate of interest, if the payment happens at the end of the year:

$\begin{array}{c}\text{Monthly rate of interest}={\left(1+\text{Rate of return}\right)}^{\frac{1}{12}}-1\\ ={\left(1+0.05\right)}^{\frac{1}{12}}-1\\ =0.004074\text{ or }0.4074\%\end{array}$

Hence, the monthly rate of interest is 0.4074%.

Equation of PV (Present Value) for perpetuity to compute the monthly charge, if the payment happens at the end of the month:

$\text{Present value}=\frac{\text{Monthly charge}}{\text{Monthly rate of interest}}$

Compute the monthly charge, if the payment happens at the end of the month:

$\begin{array}{c}\text{Present value}=\frac{\text{Monthly charge}}{\text{Monthly rate of interest}}\\ \$390,000=\frac{\text{Monthly charge}}{0.004074}\\ \text{Monthly charge}=\$1,588.86\end{array}$

Hence, the monthly charge, if the payment happens at the end of the month is $1,588.86.

Equation of PV (Present Value) for perpetuity to compute the monthly charge, if the payment happens at the starting of the month:

$\text{Present value}=\text{Monthly charge}+\frac{\text{Monthly charge}}{\text{Required rate of return}}$

Compute the monthly charge, if the payment happens at the starting of the month:

$\begin{array}{c}\text{Present value}=\text{Monthly charge}+\frac{\text{Monthly charge}}{\text{Required rate of return}}\\ \text{Monthly charge}=\left(\frac{\text{Present value}\times \text{Required rate of return}}{1+\text{Required rate of return}}\right)\\ =\frac{\$390,000\times 0.004074}{1+0.004074}\\ =\$1,582.41\end{array}$

Hence, the monthly charge, if the payment happens at the beginning of the month is $1,582.41.